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THE 

American House Carpenter, 

A TREATISE 

ON THE 

ART OF BUILDING. 

COMPRISING 

STYLES OF ARCHITECTURE, STRENGTH OF MATERIALS, 



THE THEORY AND PRACTICE OF THE CONSTRUCTION OF FLOORS, FRAMED 

GIRDERS, ROOF TRUSSES, ROLLED-IRON BEAMS, TUBULAR-IRON 

GIRDERS, CAST-IRON GIRDERS, STAIRS, DOORS, 

WINDOWS, MOULDINGS, AND CORNICES; 

TOGETHER WITH 

A COMPEND OF MATHEMATICS. 



A MANUAL FOR THE PRACTICAL USE OF 

ARCHITECTS, CARPENTERS, STAIR-BUILDERS. 

AND OTHERS. 



pU J EiaHTH EDITION, 

^-ly REWRITTEN AND ENLARGED. 

BY 



R. G. HATFIELD, ARCHITECT, 



LATK FELLOW OF THE AMERICAN INSTITUTE OF ARCHITECTS, MEMBER OF THE AMBRICAM 
SOCIETY OF CIVIL ENGINEERS, ETC, 

AUTHOR OF "TRANSVERSE STRAINS." 

Edited by O. P. HATFIELD, F.A.I.A., Architect. 

^ - iof^,^- 

NEW YORK: 
JOHN WILEY & SONS, 15 ASTOR PLACE 

isio. 






Copyright, 1880, 
Bv THE ESTATE OF R. G. HATFIELD. 



Press of Johk A. Gray, Act., 
18 Jacob Street, 

NEW YORK. 



d 




/ 



70? 



f 



PREFACE 



Since the publication of the first edition of this work, six subsequent 
editions have been issued ; but, although from time to time many additions 
to its pages and revisions of its subject-matter have been made, still its sev- 
eral issues have always been printed substantially from the original stereotype 
plates. In this edition^ however, the book has been extensively remodelled 
and expanded, the greater portion of it rewritten, and the whole put in a new 
dress by being newly set up in type uniform in style with that of the late 
author's recent work, Transverse Strains. To this revision — a labor of love to 
him — he devoted all the time he could spare from his other pressing engage- 
ments for a year or more, and by close and arduous application brought the 
book to a successful termination, notwithstanding the engrossing nature of 
his customary business avocations. Although essentially an elementary work, 
and intended originally for a class of minds not generally favored with oppor- 
tunities for securing a very extended form of education, either in the store of 
information acquired or in the discipline of mind which culture confers, still 
it has been his aim to embody in its pages so complete and exhaustive a treat- 
ment of the various subjects discussed, and so practical and useful a collection 
of data and the rules governing their application, as to make it also not un- 
worthy the attention of those who have been more highly favored in that 
respect. 

In all the various trades connected with building it is the intelligent 
workman that commands the greatest respect, and who receives in all cases 
the highest remuneration. As apprentice, journeyman, and master-builder, 
his course is upward and onward, and success crowns his eflforts in all that be 
undertakes. There is a kind of freemasonry in the very air that surrounds 
the skilful, intelligent man, that gives him a pass at once into the appreciation 
and recognition of all those whose regard is valuable. We admire and respect 
the plodding toil of the honest, patient laborer, whose humble task may tax 
his muscles though not his mind, but we yield a deeper homage to the skilful 
hand and tutored eye that accomplish wonders in art and science through per- 
severance in aspiring studies. It was to excite in the minds of workmen like 
these an ambition to excel in their calling, and to point out to them the surest 
path to that consummation, that the preparation of this volume was under- 
taken ; that all its tendencies are in that direction, and that it cannot well fail 



11 PREFACE. 

of its purpose when judiciously used, must be the conviction of all who will 
take the trouble to examine its pages. 

In the first part of the book matters more particularly relating to building 
are treated of. The first section is in the nature of an introduction, serving 
by its historical references to excite an interest in the general subject, while 
in the second are presented the methods of erecting edifices in accordance 
with the acknowledged principles of sound construction. In the remaining 
sections of Part I. the several well-defined branches of house-building, as 
stairs, doors and windows, etc., are illustrated and explained. In the second 
part the more useful rules and simple problems of mathematics are reduced 
to an easily acquired form, and adapted to the necessities of the ordinary 
workman. By studying the latter, the young mechanic may not only improve 
and strengthen his mind, but grow more self-reliant daily, demonstrating in 
his own experience that scientific knowledge gives power. By carefully 
studying this part of the book he will see how easy it is to acquire the knowl- 
edge of solving problems by signs and symbols, commonly called Algebra 
(although looked upon by the uninitiated as almost incomprehensible), and 
thus find it easy to understand all the illustrations of the various subjects 
wherein those condensed forms of expression are used. Useful problems in 
geometry, described in simple language, and hints upon the subject of draw- 
ing and shading, are also to be found in Part II. A glossary of architectural 
terms and many useful tables are provided in the Appendix, and finally, an 
Index is added to aid in referring to special subjects. The full-plate illustra- 
tions are inserted to make it attractive to the general reader, and at the same 
time to serve as explanatory of the historical portion of the volume. 

It will not be denied that the class of information herein furnished is one 
of the most instructive and useful that can be presented to the practical mind 
of a workingman, or to any mind engaged in mechanical pursuits. The im- 
press stamped upon it by the author's peculiar line of study is not to be 
effaced, but this has given it characteristics of originality and strength not 
to be found in a mere compilation. 

THE EDITOR. 

New York, 31 Pine Street, 
January 6, 1S80. 



CONTENTS, 



For a Table of Contents more in detail, see p. 613, and for Index, see p. 657, 



PART I. 

PAGE. 

Section I. — Architecture 5 

II. — Construction 57 

III. — Stairs 240 

IV. — Doors and Windows 315 

V, — Mouldings and Cornices 323 



PART IT 

Section VI. — Geometry 347 

VII. — Ratio or Proportion 366 

VIII. — Fractions 378 

" IX. — Algebra 392 

X. — Polygons ..., 439 

XL— The Circle 468 

'• XII.— The Ellipse 481 

XIII.— The Parabola 492 

XIV. — Trigonometry 510 

" XV. — Drawing 536 

XVI. — Practical Geometry 544 

" XVII.— Shadows 596 

Table of Contents in detail 613 . 



IV CONTENTS. 

APPENDIX. 

Glossary 627 

Table of Squares, Cubes and Roots , . . . 638 

Rules for the Reduction of Decimals 647 

Table of Circles 649 

Table showing the Capacity of Wells, Cisterns, etc 653 

Table of the Weights of Materials 654 

Index 657 



PART I 



SECTION I.— ARCHITECTURE, 



Art. I Buildinir Defined. — Building and Architecture 

are technical terms by some thought to be synonymous ; 
but there is a distinction. Architecture has been defined to 
be— *'the art of building;" but more correctly it is — ''the 
art of designing and constructing buildings, in accordance 
with such principles as constitute stability, utility, and 
beauty." The literal signification of the Greek word arc/it- 
tecton, from which the word architect is derived, is chief- 
carpenter; and the architect who designs and builds well 
may truly be considered the chief builder. Of the three 
classes into which architecture has been divided — viz.. Civil, 
Military, and Naval — the first is that which refers to the 
construction of edifices known as dwellings, churches, and 
other public buildings, bridges, etc., for the accommodation 
of civilized man — and is the subject of the remarks which 
follow. 

2. — Antique Buildings; Tower of Babel. — Building ia 
one of the most ancient of the arts : the Scriptures inform 
us of its existence at a very early period. Cain, the son of 
Adam, '' builded a city, and called the name of the city 
after the name of his son, Enoch ;" but of the peculiar style 
or manner of building we are not informed. It is presumed 
that it was not remarkable for beauty, but that utility and 
perhaps stabihty were its characteristics. Soon after the 
deluge — that memorable event, which removed from ex- 
istence all traces of the works of man — the Tower of Babel 



6 ARCHITECTURE. 

was commenced. This was a work of such magnitude that 
the gathering of the materials, according to some writers, 
occupied three years; the period from its commencement 
until the work was abandoned was twenty-two years ; and 
the bricks were like blocks of stone, being twenty feet long, 
fifteen broad, and seven thick. Learned men have given it 
as their opinion that the tower in the temple of Belus at 
Babylon was the same as that which in the Scriptures is 
called the Tower of Babel. The tower of the temple of 
Belus was square at its base, each side measuring one 
furlong, and consequently half a mile in circumference. Its 
form was that of a pyramid, and its height was 660 feet. It 
had a winding passage on the outside from the base to the 
summit, which was wide enough for two carriages. 

3, — Ancient Cities and Monuments. — Historical accounts 
of ancient cities, such as Babylon, Palmyra, and Nineveh of 
the Assyrians ; Sidon, Tyre, Aradus, and Serepta of the 
Phoenicians ; and Jerusalem, with its splendid temple, of 
the Israelites — show that architecture among them had 
made great advances. Ancient monuments of the art are 
found also among other nations ; the subterraneous temples 
of the Hindoos upon the islands Elephanta and Salsetta ; 
the ruins of Persepolis in Persia ; pyramids, obelisks, tem- 
ples, palaces, and sepulchres in Egypt — all prove that the 
architects of those early times were possessed of skill and 
judgment highly cultivated. The principal characteristics 
of their works are gigantic dimensions, immovable solidity, 
and, in some instances, harmonious splendor. The extra- 
ordinary size of some is illustrated in the pyramids of Egypt. 
The largest of these stands not far from the city of Cairo : 
its base, which is square, covers about ii|- acres, and its 
height is nearly 500 feet. The stones of which it is built 
are immense — the smallest being full thirty feet long. 

4. — Architecture in Oreece. — Among the Greeks, archi- 
tecture was cultivated as a fine art. Dignity and grace 
were added to stability and magnificence. In the Doric 
order, their first style of building, this is fully exemplified. 
Phidias, Ictinus, and Calicrates are spoken of as masters in 



GRECIAN AND ROMAN BUILDINGS. 7 

the art at this period : the encouragement and support of 
Pericles stimulated them to a noble emulation. The beauti- 
ful temple of Minerva, called the Parthenon, erected upon 
the acropolis of Athens, the Propyleum, the Odeum, and 
others, were lasting monuments of their success. The Ionic 
and Corinthian orders were added to the Doric, and many 
magnificent edifices arose. These exemplified, in their 
chaste proportions, the elegant refinement of Grecian taste. 
Improvement in Grecian architecture continued to advance 
until perfection seems to have been attained. The speci- 
mens which have been partially preserved exhibit a com- 
bination of elegant proportion, dignified simplicity, and 
majestic grandeur. Architecture among the Greeks was at 
the height of its glory at the period immediately preceding 
the Peloponnesian war ; after which the art declined. An 
excess of enrichment succeeded its former simple grandeur ; 
yet a strict regularity was maintained amid the profusion of 
ornament. After the death of Alexander, 323 B.C., a love 
of gaudy splendor increased : the consequent decline of the 
art was visible, and the Greeks afterwards paid but little 
attention to the science. 

5. — Arcliitecture in Rome. — While the Greeks illustrated 
their knowledge of architecture in the erection of their 
temples and other public buildings, the Romans gave their 
attention to the science in the construction of the many 
aqueducts and sewers with which Rome abounded ; build- 
ing no such splendid edifices as adorned Athens, Corinth, 
and Ephesus, until about 200 years B.C., when their inter- 
course with the Greeks became more extended. Grecian 
architecture was introduced into Rome by Sylla ; by Avhom, 
as also by Marius and Csesar, many large edifices were 
erected in various cities of Italy. But under Caesar Augus- 
tus, at about the beginning of the Christian era, the art 
arose to the greatest perfection it ever attained in Italy. 
Under his patronage Grecian artists were encouraged, and 
many emigrated to Rome. It was at about this time that 
Solomon's temple at Jerusalem was rebuilt by Herod — a 
Roman. This was 46 years in the erection, and was most 
probably of the Grecian style of building — perhaps of the 



8 ARCHITECTURE. 

Corinthian order. Some of the stones of which it was built 
were 46 feet long, 21 feet high, and 14 thick; and others 
were of the astonishing length of 82 feet. The porch rose 
to a great height ; the whole being built of white marble 
exquisitely polished. This is the building concerning which 
it was remarked : '' Master, see what manner of stones, and 
what buildings are here." For the construction of private 
habitations also, finished artists were employed by the 
Romans : their dwellings being often built with the finest 
marble, and their villas splendidly adorned. After Augus- 
tus, his successors continued to beautify the city, until the 
reign of Constantine, who, having removed the imperial 
residence to Constantinople, neglected to add to the splen- 
dor of Rome ; and the art, in consequence, soon fell from its 
high excellence. 

6, — Rome and Greece. — Thus Rome was indebted to 
Greece for her knowledge of architecture — not only for the 
knowledge of its principles, but also for many of the best 
buildings themselves ; these having been originally erected 
in Greece, and stolen by the unprincipled conquerors — 
taken down and removed to Rome. Greece was thus rob- 
bed of her best monuments of architecture. Touched by 
the Romans, Grecian architecture lost much of its elegance 
and dignity. The Romans, though justly celebrated for 
their scientific knowledge as displayed in the construction 
of their various edifices, were not capable of appreciating 
the simple grandeur, the refined elegance of the Grecian 
style ; but sought to improve upon it by the addition of 
luxurious enrichment, and thus deprived it of true elegance. 
*In the days of Nero, whose palace of gold is so celebrated, 
buildings were lavishly adorned. Adrian did much to en- 
courage the art ; but not satisfied with the simplicity of the 
Grecian style, the artists of his time aimed at inventing new 
ones, and added to the already redundant embellishments of 
the previous age. Hence the origin of the pedestal, the 
great variety of intricate ornaments, the convex frieze, the 
round and the open pediments, etc. The rage for luxury 
continued until Alexander Scverus, who made some im- 



p 



I 




CATHEDRAL OF NOTRE DAME, PARIS. i^^ f^ 



THE GOTHS AND VANDALS. 9 

provement ; but very soon after his reign the art began 
rapidly to decUne, as particularly evidenced in the mean 
and trifling character of the ornaments. 

7. — Arcliitecturc Debased. — The Goths and Vandals 
overran Italy, Greece, Asia, and Africa, destroying most 
of their works of ancient architecture. Cultivating no art 
but that of war, these savage hordes could not be expected i 
to take any interest in the beautiful forms and proportions | 
of their habitations. From this time architecture assumed 
an entirely different aspect. The celebrated styles of Greece 
were unappreciated and forgotten ; and modern architec- 
ture made its first appearance on the stage of existence. 
The Goths, in their conquering invasions, gradually ex- 
tended it over Italy, France, Spain, Portugal, and Ger- 
many, into England. From the reign of Galienus may be 
reckoned the total extinction of the arts among the Romans. 
From this time until the sixth or seventh century, architec- 
ture was almost entirely neglected. The buildings which 
were erected during this suspension of the arts were very 
rude. Being constructed of the fragments of the edifices 
which had been demolished by the Visigoths in their unre- 
strained fury, and the builders being destitute of a proper 
knowledge of architecture, many sad blunders and exten- 
sive patch-work might have been seen in their construction 
— entablatures inverted, columns standing on their wrong 
ends, and other ridiculous arrangements characterized their 
clumsy work. The vast number of columns which the ruins 
around them afforded they used as piers in the construction 
of arcades — which by some is thought, after having passed 
through various changes, to have been the origin of the 
plan of the Gothic cathedral. Buildings generally, which 
are not of the classical styles, and w^hich were erected after 
the fall of the Roman empire, have by some been indiscrim- 
inately included under the term Gothic. But the changes 
which architecture underwent during the Mediaeval age 
show that there were then several distinct modes of building. 

8. — Tlic Ostrogoths. — Theodoric, a friend of the arts, 
who reigned in Italy from A.D. 493 to 525, endeavored to 



lO ARCHITECTURE. 

restore and preserve some of the ancient buildings ; and 
erected others, the ruins of which are still seen at Verona 
and Ravenna. Simplicity and strength are the character- 
istics of the structures erected by him ; they are, however, 
devoid of grandeur and elegance, or fine proportions. 
These are properly of the Gothic style ; by some called 
the old Gothic, to distinguish it from the pointed Gothic. 

9. — Tlie Eiombards, who ruled in Italy from A.D. 568, 
had no taste for architecture nor respect for antiquities. 
Accordingly, they pulled down the splendid monuments of 
classic architecture which they found standing, and erected 
in their stead huge buildings of stone which were greatly 
destitute of proportion, elegance, or utility — their charac- 
teristics being scarcely anything more than stability and 
immensity combined with ornaments of a puerile character. 
Their churches were decorated with rows of small columns 
along the cornice of the pediment, small doors and win- 
dows with circular heads, roofs supported by arches having 
arched buttresses to resist their thrust, and a lavish display 
of incongruous ornaments. This kind of architecture is 
called the Lombard style, and was employed in the seventh 
century in Pavia, the chief city of the Lombards ; at which 
city, as also at many other places, a great many edifices 
Avere erected in accordance with its peculiar forms. 

10. — TSsc Myzantine Architects, of Byzantium, Constan- 
tinople, erected many spacious edifices ; among which are 
included the cathedrals of Bamberg, Worms, and Mentz, 
and the most ancient part of the minster at Strassburg ; in 
all of these they combined the classic styles with the crude 
Lombardian. This style is called the Lombard-Byzantine. 
To the last style there were afterwards added cupolas sim- 
ilar to those used in the East, together with numerous slen- 
der pillars with elaborate capitals, and the many minarets 
which are the characteristics of the proper Byzantine, or 
Oriental style. 

M. — The moors. — When the Arabs and Moors destroyed 
the kingdom of the Goths, the arts and sciences were mostly 








MOSQUE AT CAIRO. 



THE MEDIEVAL STYLES. II 

in possession of the Musselmen-conquerors ; at which time 
there were three kinds of architecture practised ; viz. : the 
Arabian, the Moorish, and the Lombardian. The Arabian 
style was formed from Greek models, having circular arches 
added, and towers which terminated with globes and mina- 
rets. The Moorish is very similar to the Arabian, being 
distinguished from it by arches in the form of a horseshoe. 
It originated in Spain in the erection of buildings with the 
ruins of Roman architecture, and is seen in all its splendor 
in the ancient palace of the Mohammedan monarchs at 
Grenada, called the Alhambra, or rcd-Jwtise. The style which 
was originated by the Visigoths in Spain by a combination 
of the Arabian and Moorish styles, was introduced by Charle- 
magne into Germany. On account of the changes and im- 
provements it there underwent, it was, at about the 13th or 
14th century, termed the German or romantic style. It is ex- 
hibited in great perfection in the towers of the minster of 
Strassburg, the cathedral of Cologne and other edifices. 
The most remarkable features of this lofty and aspiring style 
are the lancet or pointed arch, clustered pillars, lofty towers, 
and flying buttresses. It was principally employed in eccle- 
siastical architecture, and in this capacity introduced into 
France, Italy, Spain, and England. 

12. — T&e Architecture of Etigland : is divided into the 
Norman^ the Early-EnglisJiy the Decorated, and the Pcrpe^idic- 
ular styles. The Norman is principally distinguished by 
the character of its ornaments — the chevron, or zigzag, being 
the most common. Buildings in this style were erected in 
the 1 2th century. The Early-English is celebrated for the 
beauty of its edifices, the chaste simplicity and purity of 
design which they display, and the peculiarly graceful char- 
acter of its foliage. This style is of the 13th century. The 
Decorated style, as its name implies, is characterized by a 
great profusion of enrichment, which consists principally of 
the crocket, or feathered-ornament, and ball-flower. It was 
mostly in use in the 14th century. The Perpendicular style, 
which dates from the 15th century, is distinguished by its 
high towers, and parapets surmounted with spires similar in 
number and grouping to oriental minarets. 



12 ARCHITECTURE. 

13. — Architectwre Progressive. — The styles erroneously 
termed GotJiic were distinguished by peculiar characteris- 
tics as well as by different names. The first symptoms of a 
desire to return to a pure style in architecture, after the 
ruin caused by the Goths, was manifested in the character 
of the art as displayed in the church of St. Sophia at Con- 
stantinople, which was erected by Justinian in the 6th 
century. The church of St. Mark at Venice, which arose 
in the loth or nth century, is a most remarkable building; 
a compound of many of the forms of ancient architecture. 
The cathedral at Pisa, a wonderful structure for the age, 
was erected by a Grecian architect in ioi6. The marble 
with which the walls of this building were faced, and of 
which the four rows of columns that support the roof are 
composed, is said to be of an excellent character. The 
Campanile, or leaning-tower as it is usually called, was 
erected near the cathedral in the I2th century. Its inclina- 
tion is generally supposed to have arisen from a poor foun- 
dation ; although by some it is said to have been thus con- 
structed originally, in order to inspire in the minds of the 
beholder sensations of sublimity and awe. In the 13th cen- 
tury, the science in Italy was slowly progressing ; many fine 
churches were erected, the style of which displayed a de- 
cided advance in the progress towards pure classical archi- 
tecture. In other parts of Europe, the Gothic, or pointed 
style was prevalent. The cathedral at Strassburg, designed 
by Irwin Steinbeck, was erected in the 13th and 14th cen- 
turies. In France and England during the 14th century, 
many very superior edifices were erected in this style. 

14. — AFclaitecturc in Italy. — In the 14th and 15th cen- 
turies, architecture in Italy was greatly revived. The mas- 
ters began to study the remains of ancient Roman edifices ; 
and many splendid buildings were erected, which displayed 
a purer taste in the science. Among others, St. Peter's of 
Rome, which was built about this time, is a lasting monu- 
ment of the architectural skill of the age. Giocondo, Mi- 
chael Angelo, Palladio, Vignola, and other celebrated archi- 
tects, each in their turn, did much to restore the art to its 




INTERIOR OF ST. SOPHIA. CONSTANTINOPLE. 



I 



ORIGIN OF STYLES. 1 3 

former excellence. In the edifices which were erected under 
their direction, however, it is plainly to be seen that they 
studied not from the pure models of Greece, but from the 
remains of the deteriorated architecture of Rome. The 
high pedestal, the coupled columns, the rounded pediment, 
the many curved-and-twisted enrichments, and the convex 
frieze, were unknown to pure Grecian architecture. Yet 
their efforts were serviceable in correcting, to a good de- 
gree, the very impure taste that had prevailed since the over- 
throw of the Roman empire. 

(5. — The Rciiai§§ance. — The Italian masters and numer- 
ous artists w^ho had visited Italy for the purpose, spread the 
Roman style over various countries of Europe ; which was 
gradually received into favor in place of the pointed Gothic. 
This fell into disuse ; although it has of late years been 
again cultivated. It requires a building of great magnitude 
and complexity for a perfect display of its beauties. In 
America, the pure Grecian style Avas at first more or less 
studied ; and perhaps the simplicity of its principles would 
be better adapted to a republican country than the more 
intricate mediasval styles ; yet these, during the last quarter 
of a century, have been extensively studied, and now wholly 
supersede the Grecian styles. 

(6. — Styles of Arcliitecture. — It is generally acknowl- 
edged that the various styles in architecture were the results 
of necessity, and originated in accordance with the different 
pursuits of the early inhabitants of the earth ; and were 
brought by their descendants to their present state of per- 
fection, through the propensity for imitation and desire of 
emulation which are found more or less among all nations. 
Those that followed agricultural pursuits, from being em- 
ploA^ed constantly upon the same piece of land, needed a 
permanent residence, and the wooden Jiut was the offspring 
of their wants ; while the shepherd, who followed his flocks 
and was compelled to traverse large tracts of country for 
pasture, found the tent to be the most portable habitation ; 
again, the man devoted to hunting and fishing — an idle and 
vagabond way of living — is naturally supposed to have been 



L 



14 ARCHITECTURE. 

content with the cavern as a place of shelter. The latter is 
said to have been the origin of the Egyptian style ; while 
the curved roof of Chinese structures gives a strong indica- 
tion of their having had the tent for their model ; and the 
simplicity of the original style of the Greeks (the Doric) 
shows quite conclusively, as is generally conceded, that its 
original was of wood. The pointed, or ecclesiastical style, 
is said to have originated in an attempt to imitate the bower, 
or grove of trees, in which the ancients performed their idol- 
worship. But it is more probably the result of repeated 
scientific attempts to secure real strength with apparent 
lightness ; thus giving a graceful, aspiring effect. 

17. — Orders: or styles, in architecture are numerous; 
and a knowledge of the peculiarities of each is important to 
the student in the art. An order, in architecture, is com- 
posed of three principal parts, viz. : the Stylobate, the Col- 
umn, and the Entablature. This appertains chiefly to the 
classic styles. 

18. — The Stylobate: is the substructure, or basement, 
upon which the columns of an order are arranged. In 
Roman architecture — especially in the interior of an edi- 
fice — it frequently occurs that each column has a separate 
substructure ; this is called a pedestal. If possible, the ped- 
estal should be avoided in all cases ; because it gives to the 
column the appearance of having been originally designed 
for a small building, and afterwards pieced out to make it 
long enough for a larger one. 

19. — The Column: is composed of the base, shaft, and 
capital. 

20. — The Entablature : above and supported b}^ the 
columns, is horizontal ; and is composed of the architrave, 
frieze, and cornice. These principal parts are again divided 
into various members and mouldings. 

21. — The Base: of a column is so called from basis, a 
foundation or footing. 




INTERIOR OF ST. i T C > I 5 <' i, I' V il ^. 



PARTS OF AN ORDER. 1 5 

22. — Tlie Shaft: the upright part of a column standing 
upon the base and crowned with the capital, is from shafto, 
to dig — in the manner of a well, whose inside is not unlike 
the form of a column. 

23. — The Capital: from kephale or caput, the head, is the 
uppermost and crowning part of the column. 

24. — The Architrave : from arcJii, chief or principal, 
and trabs, a beam, is that part of the entablature which lies 
in immediate connection with the column. 

25. — The Frieze: irom. fibron, a fringe or border, is that 
part of the entablature which is immediately above the 
architrave and beneath the cornice. It was called by some 
of the ancients zopJwriis, because it was usually enriched 
with sculptured animals. 

26. — The Cornice: from corona, a crown, is the upper 
and projecting part of the entablature — being also the upper- 
most and crowning part of the whole order. 

27- — The Pccliinent : above the entablature, is the tri- 
angular portion which is formed by the mclined edges of 
the roof at the end of the building. In Gothic architecture, 
the pediment is called a gable. 

28. — The Tyinpaiium : is the perpendicular triangular 
surface which is enclosed by the cornice of the pediment. 

29. — The Attic: is a small order, consisting of pilasters 
and entablature, raised above a larger order, instead of a 
pediment. An attic story is the upper story, its windows 
being usually square. 

30. — Proportioii§ in an Order. — An order has its several 
members proportioned to one another by a scale of 60 equal 
parts, which are called minutes. If the height of buildings 
were always the same, the scale of equal parts would be a 
hxed quantity — an exact number of feet and inches. But as 
buildings are erected of different heights, the column and 



1 6 ARCHITECTURE, 

its accompaniments are required to be of different dimen- 
sions. To ascertain the scale of equal parts, it is necessary 
to know the height to which the Avhole order is to be 
erected. This must be divided by 'the number of diameters 
which is directed for the order under consideration. Then 
the quotient obtained by such division is the length of the 
scale of equal parts — and is, also, the diameter of the column 
next above the base. For instance, in the Grecian Doric 
order the whole height, including column and entablature, 
is 8 diameters. Suppose now it were desirable to construct 
an example of this order, forty feet high. Then 40 feet 
divided by 8 gives 5 feet for the length of the scale ; and 
this being divided by 60, the scale is completed. The up- 
right columns of figures, marked H and P, by the side of 
the drawings illustrating the orders, designate the height 
and the projection of the members. The projection of each 
member is reckoned from a line passing through the axis of 
the column, and extending above it to the top of the entab- 
lature. The figures represent minutes, or 6oths, of the 
major diameter of the shaft of the column. 

31. — Grecian Styles.— The original method of building 
among the Greeks was in what is called the Doric order : 
to this were afterwards added the Ionic and the Corinthian. 
These three were the only styles known among them. Each 
is distinguished from the other two by not only a peculiar- 
ity of some one or more of its principal parts, but also by a 
particular destination. The character of the Doric is robust, 
manly, and Herculean-like ; that of the Ionic is more deli- 
cate, feminine, matronly ; while that of the Corinthian is 
extremely delicate, youthful, and virgin-like. However 
they may differ in their general character, they are alike 
famous for grace and diguii}', elegance and grandeur, to a 
high degree of perfection. 

32. — The l>oric Order: {Fig^, 2,) is so ancient that its 
origin is unknown — although some have pretended to have 
discovered it. But the most general opinion is, that it is 
an improvement upon the original wooden buildings of the 




^^iiiiiiiiiiiE:: aiiiiiiiii 



FA^XIFUL ORICIX OF THE DORIC. 



17 



Grecians. These no doubt were very rude, and perhaps 
not unhke the following figure. 




Fig. I. — Supposed Origin of Doric Temple. 



The trunks of trees, set perpendicularly to support the 
roof, may be taken for columns ; the tree laid upon the 
tops of the perpendicular ones, the architrave ; the ends 
of the cross-beams which rest upon the architrave, the 
triglyphs ; the tree laid on the cross-beams as a support for 
the ends of the rafters, the bed-moulding of the cornice ; the 
ends of the rafters which project beyond the bed-moulding, 
the mutules; and perhaps the projection of the roof in 
front, to screen the entrance from the weather, gave origin 
to the portico. 

The peculiarities of the Doric order are the triglyphs — 
those parts of the frieze which have perpendicular channels 
cut in their surface ; the absence of a base to the column — 
as also of fillets between the flutings of the column ; and the 
plainness of the capital. The triglyphs should be so dis- 
posed that the width of the m.etopes — the space between 
the triglyphs — shall be equal to their height. 



33. — The Intcrcolumniation : or space between the col- 
umns, is regulated by placing the centres of the columns 
under the centres of the triglyphs — except at the angle ot 
the building ; where, as may be seen in Fig. 2, one edge of 



i8 



ARCHITECTURE. 




-i-LJ— i_n-T-rT^ T 



.n n-n-i-uT 



jn_n_nLJ.^-i-j_i 



r^ /^^ 



28 



3 1 30 



30 



J LJ U 



2n 



22J 



s; 



n 



^-c^j 



vli L-J lJ LJ 




1 I f n 

— ' 



Fig, 2. — Grecian Doric. 



k 



PECULIARITIES OF THE DORIC. 1 9 

the triglyph must be over the centre of the column."^' 
Where the columns are so disposed that one of them stands 
beneath every other triglyph, the arrangement is called 
vtono-triglyph and is most common. When a column is 
placed beneath every third triglyph, the arrangement is 
called diastyle ; and when beneath every fourth, arceostyle. 
This last style is the worst, and is seldom adopted. 

34.— The Doric Order: is suitable for buildings that 
are destined for national purposes, for banking-houses, etc. 
Its appearance, though massive and grand, is nevertheless 
rich and graceful. The Patent Office at Washington, and 
the Treasury at New York, are good specimens of this 
order. 

35- — The Ionic Order. {Fig. 3.) — The Doric Avas for 
some time the only order in use among the Greeks. They 
gave their attention to the cultivation of it, until perfection 
seems to have been attained. Their temples were the prin- 

* Grecian Doric Order. When the -width to be occupied by the whole front 
is limited, to determine the diameter of the column. 

The relation between the parts may be expressed thus : 
_ 60 <7 

^ ~ ^7(^+ c) + (60 — c) 

Where a equals the width in feet occupied by the columns, and their inter- 
columniations taken collectively, measured at the base ; b equals the width 
of the metope, in minutes ; c equals the width of the triglyphs in minutes ; d 
equals the number of metopes, and x equals the diameter in feet. 

Example. — A front of six columns — hexastyle — 61 feet wide ; the frieze 
having one triglyph over each intcrcolumniation, or mono-triglyph. In this 
case, there being five intercolumniations and two metopes over each, therefore 
there are 5 x 2 = 10 metopes. Let the metope equal 42 minutes and the 
triglyph equal 28. Then a = 61 ; (^ = 42 ; <: = 28 ; and d =z io\ and the formula 
above becomes 

60 X 61 60 X 61 3660 
— — =5 feet = the diameter 



10(42 + 28) + (60 — 28) 10 X 70 + 32 732 
required. 

Example. — An octastyle front, 8 columns, 1S4 feet wide, three metopes 
over each intercolumniation, 21 in all, and the metope and triglyph 42 and 
28, as before. Then 

60 X 184 11040 -. , , J. , J 

^ — — ;; a\ 72 ^ — • = 7-35t5 0!Z feet = the diameter required. 

21 (42 + 28) + (60 — 28) 1502 ' -'-'i^o-f n 



20 ARCHITECTURE. 

cipal objects vipon which their skill in the art was displayed ; 
and as the Doric order seems to have been well fitted, by its 
massive proportions, to represent the character of their 
male deities rather than the female, there seems to have 
been a necessity for another style which should be emble- 
matical of feminine graces, and with which they might 
decorate such temples as were dedicated to the goddesses. 
Hence the origin of the Ionic order. This was invented, 
according to historians, by Hermogenes of Alabanda ; and 
he being a native of Caria, then in the possession of the 
lonians, the order was called the Ionic. 

The distinguishing features of this order are the volutes 
or spirals of the capital ; and the dentils among the bed- 
mouldings of the cornice: although in some instances 
dentils are wanting. The volutes are said to have been 
designed as a representation of curls of hair on the head of 
a matron, of whom the whole column is taken as a sem- 
blance. 

The Ionic order is appropriate for churches, colleges, 
seminaries, libraries, all edifices dedicated to literature and 
the arts, and all places of peace and tranquillity. The front 
of the Custom-House, New York City, is a good specimen 
of this order. 

36. — The Intcrcolumniation : of this and the other 
orders — both Roman and Grecian, with the exception of 
the Doric — are distinguished as follows. When the interval 
is one and a half diameters, it is cdXi^^ pycno style, or columns 
thick-set; when two diameters, systyle ; when two and a 
quarter diameters, t'?/j/j/^/ when three diameters, <^/rt'j/>/^ / 
and when more than three diameters, arcEostyle, or columns 
thin-set. In all the orders, when there are four columns in 
one row, the arrangement is called tetrastyle ; when there 
are six in a row, Jiexasiyle ; and when eight, octastyle. 

37. — To De§cribc the Ionic Volute.— Draw a perpen- 
dicular from ^ to ^ {Fig. 4), and make a s equal to 20 min. 
or to -4- of the whole height, a c ; draw s 2X right angles to 
s a, and equal to i^ min. ; upon o, with 2\ min. for radius. 



'ROPORTIONS OF GRECIAN IONIC. 




Fig. 3. — Grecian Ionic. 



'22 



ARCHITECTURE. 



describe the eye of the volute ; about o, the centre of the 
eye, draw the square, r t i 2, with sides equal to half the 
diameter of the eye, viz. 2^ min., and divide it into 144 equal 
parts, as shown at Fig. 5. The several centres in rotation are 
at the angles formed by the heavy lines, as figured, i, 2, 3, 
4, 5, 6, etc. The position of these angles is determined by 
commencing at the point, i, and making each heavy line one 
part less in length than the preceding one. No. i is the 




Fig. 4. — Ionic Volute. 



THE IONIC VOLUTE. 



23 



centre lor the arc a b {Fig, 4 ;) 2 is the centre for the arc 
b c ; and so on to the last. The inside spiral line is to be 
described from the centres, x, x, x, etc. {Fig. 5), being the 
centre of the first small square towards the middle of the 
eye from the centre for the outside arc. The breadth of the 
fillet at a j is to be made equal to 2-^^ min. This is for a spiral 
of three revolutions ; but one of any number of revolutions, 
as 4 or 6, may be drawn, by dividing of {Fig. 5) into a cor- 
responding number of equal parts. Then divide the part 
nearest the centre, <?, into two parts, as at h ; join and i, 
also and 2 ; draw // 3 parallel to ^ i, and h 4 parallel to o 




Fig. 5. — Eye of Volute. 

2 ; then the lines ^ i, ^ 2, /^ 3, // 4 will determine the length 
of the heavy lines, and the place of the centres. (See Art. 
288.) 



38. — The Corinthian Order : {Fig. 7,) is in general like 
the Ionic, though the proportions are lighter. The Corin- 
thian displays a more airy elegance, a richer appearance ; 
but its distinguishing feature is its beautiful capital. This 
is generally supposed to have had its origin in the capitals 



24 



ARCHITECTURE. 




,aJUH 



Fig. 6. 



of the columns of Egyptian temples, which, though not ap- 
proaching it in elegance, have j^et a similarity of form with 
the Corinthian. The oft-repeated story of its origin which 

is told by Vitruvius — an architect 
who flourished in Rome in the days 
of Augustus Csesar — though pretty 
generally considered to be fabu- 
lous, is nevertheless worthy of be- 
ing again recited. It is this : A 
young lady of Corinth was sick, and 
finally died. Her nurse gathered 
into a deep basket such trinkets and 
keepsakes as the lady had been 
fond of Avhen alive, and placed them upon her grave, cover- 
ing the basket with a flat stone or tile, that its contents 
might not be disturbed. The basket was placed accident- 
ally upon the stem of an acanthus plant, which, shooting 
forth, enclosed the basket with its foliage, some of which, 
reaching the tile, turned gracefully over in the form of a 
volute. 

A celebrated sculptor, Calimachus, saw the basket thus 
decorated, and from the hint which it suggested conceived 
and constructed a capital for a column. This was called 
Corinthian, from the fact that it was invented and first made 
use of at Corinth. 

The Corinthian being the gayest, the richest, the most 
lovely of all the orders, it is appropriate for edifices which 
are dedicated to amusement, banqueting, and festivity — for 
all places where delicacy, gayety, and splendor are desir- 
able. 



39. — Persians and Caryatides. — In addition to the three 
regular orders of architecture, it was customary among the 
Greeks and other nations to employ representations of the 
human form, instead of columns, to support entablatures : 
these were called Persians and Caryatides. 



40. — Persians : are statues of men, and are so called in 
commemoration of a victory gained over the Persians by 
Pausanias. The Persian prisoners were brought to Athens 



PROPORTIONS OF GRECIAN CORINTHIAN. 



25 






n 





If-" 

m 






Jy. 


A'^ 



1 



Fig. 7. — Grecian Corinthian. 



26 ARCHITECTURE. 

and condemned to abject slavery ; and in order to represent 
them in the lowest state of servitude and degradation, the 
statues were loaded with the heaviest entablature, the Doric. 

4-1. — Caryatides: are statues of women dressed in long 
robes after the Asiatic manner. Their origin is as follows: 
In a war between the Greeks and the Caryans, the latter 
Avere totally vanquished, their male population extinguished, 
and their females carried to Athens. To perpetuate the 
memory of this event, statues of females, having the form 
and dress of the Caryans, were erected, and crowned with 
the Ionic or Corinthian entablature. The caryatides were 
generally formed of about the human size, but the persians 
much larger, in order to produce the greater awe and 
astonishment in the beholder. The entablatures were pro- 
portioned to a statue in like manner as to a column, of the 
same height. 

These semblances of slavery have been in frequent use 
among moderns as well as ancients ; and, as a relief from 
the stateliness and formality of the regular orders, are capa- 
ble of forming a thousand varieties ; yet in a land of liberty 
euch marks of human degradation ought not to be perpetu- 
ated. 

42. — Roman Stjlc§. — Strictl}^ speaking, Rome had no 
architecture of her own ; all she possessed was borrowed 
from other nations. Before the Romans exchanged inter- 
course with the Greeks, they possessed some edifices of 
considerable extent and merit, which were erected by archi- 
tects from Etruria ; but Rome was principally indebted to 
Greece for what she acquired of the art. Although there is 
no such thing as an architecture of Roman invention, yet 
no nation, perhaps, ever was so devoted to the cultivation 
of the art as the Roman. Whether we consider the number 
and extent of their structures, or the lavish richness and 
splendor with which they were adorned, we are compelled 
to yield to them our admiration and praise. At one time, 
under the consuls and emperors, Rome employed 400 ar- 
chitects. The public works — such as theatres, circuses, 
baths, aqueducts, etc. — were, in extent and grandeur, be- 




PORTICO OF THE ERECTHEUM, ATHEKS. 



CHANGE OF STYLES BY THE ROMANS. 2/ 

yond anything- attempted in modern times. Aqueducts 
were built to convey water from a distance of 60 miles or 
more. In the prosecution of this work rocks and mountains 
were tunnelled, and valleys bridged. vSome of the latter 
descended 200 feet below the level of the water ; and in 
passing them the canals were supported by an arcade, or 
succession of arches. Public baths are spoken of as large as 
cities, being fitted up with numerous conveniences for ex- 
ercise and amusement. Their decorations were most splen- 
did ; indeed, the exuberance of the ornaments alone was 
offensive to good taste. So overloaded with enrichments 
were the baths of Diocletian that on one occasion of public 
festivity great quantities of sculpture fell from the ceilings 
and entablatures, killing many of the people. 

43. — Grecian Orders modified by tlie Romans. — The 

orders of Greece were introduced into Rome in all their 
perfection. But the luxurious Romans, not satisfied with 
the simple elegance of their refined proportions, sought to 
improve upon them by lavish displays of ornament. They 
transformed in many instances the true elegance of the 
Grecian art into a gaudy splendor, better suited to their 
less refined taste. The Romans remodelled each of the 
orders : the Doric {Fig. 8) Avas modified by increasing the 
height of the column to 8 diameters ; by changing the 
echinus of the capital for an ovolo, or quarter-round, and 
adding an astragal and neck below it ; by placing the centre, 
instead of one edge, of the first triglyph over the centre of 
the column ; and introducing horizontal instead of inclined 
mutules in the cornice, and in some instances dispensing 
with them altogether. The Ionic was modified by diminish- 
ing the size of the volutes, and, in some specimens, intro- 
ducing a new capital in which the volutes were diagonally 
arranged {Fig. 9). This new capital has been termed modern 
Ionic. The favorite order at Rome and her colonies was 
the Corinthian {Fig. 10). But this order the Roman artists, 
in their search for novelty, subjected to many alterations- 
especially in the foliage of its capital. Into the upper part 
of this they introduced the modified Ionic capital ; thus 



ARCHITECTURE. 



combining the two in one. This change was dignified with 
the importance of an order, and received the appellation 



.66 



4" ii'i^ 

1 29. 
26 



^,.^i.^j^^ji.;^^U4'i.;^w^-A^^^ 



7 



""^liw 



L.LLLUL\ 



Uj-jUjjJj^LJj^U^^ 



^!^%^f;^^M^/^^^^;i^;$.^;^m\mMMimM^ 



Pl m 



4 4 





Fig. 8. — Roman Doric. 



of Composite, or Roman: the best specimen of which is 
found in the Arch of Titus {Fig. n). This style was not 



TROPORTIONS OF THE ROMAN IONIC. 



29 



^% 



.^f» V.v^^\AxVA..V.VA.VA.VA.X. ^V-A.AAA :v^v.^VAA.VA,VA.V,VAA.^,VA.\.\A.\.V 






.v.x..k'^ L,UL t.^^V,^V,L I.AX.V.LLU...VA AA.A-JJJ^LIJJA-.A A- 1^. X-^AA..A \li\.lK.. 



36^1 .. — ;— ■ ' 



U'^ 



'"% 



m^^^mMM^WMMfk^/^mMmA^m^/^/M^m^ 





j^^ 



€iim»illii» 




Fig. 9. — Roman Ionic. 



30 ARCHITECTURE. 

much used among the Romans themselves, and is but 
slightly appreciated now. 

4-4. — The Tu§caii Order: is said to have been intro- 
duced to the Romans by the Etruscan architects, and to 
have been the only style used in Italy before the introduc- 
tion of the Grecian orders. However this may be, its simi- 
larity to the Doric order gives strong indications of its 
having been a rude imitation of that style : this is very prob- 
able, since history informs us that the Etruscans held inter- 
course with the Greeks at a remote period. The rudeness 
of this order prevented its extensive use in Italy. All that 
is known concerning it is from Vitruvius, no remains of 
buildings in this style being found among ancient ruins. 

For mills, factories, markets, barns, stables, etc., where 
utility and strength are of more importance than beauty, 
the improved modification of this order, called the modern 
Tuscan {Fig. 12), will be useful; and its simplicity recom- 
mends it vv^here economy is desirable. 

4-5. — Egyptian Style. — The architecture of the ancient 
Eg3^ptians — to which that of the ancient Hindoos bears 
some resemblance — is characterized b}^ boldness of outline, 
solidit}^ and grandeur. The amazing labyrinths and exten- 
sive artificial lakes, the splendid palaces and gloomy ceme- 
teries, the gigantic pyramids and towering obelisks, of the 
Egyptians were Avorks of immensity and durability ; and 
their extensive remains are enduring proofs of the enlight- 
ened skill of this once-powerful but long since extinct na- 
tion. The principal features of the Egyptian style of archi- 
tecture are — uniformity of plan, never deviating from right 
lines and angles ; thick walls, having the outer surface 
slightly deviating inwardly from the perpendicular; the 
whole building low ; roof flat, composed of stones reaching 
in one piece from pier to pier, these being supported by 
enormous columns, ver}- stout in proportion to their height ; 
the shaft sometimes polygonal, having no base but with a 
great variety of handsome capitals, the foliage of these being 
of the pahn, lotus, and other leaves ; entablatures having 
simply an architrave, crowned with a huge cavetto orna- 



PROPORTIONS OF THE ROMAN CORINTHIAN. 



^I 




1 n n n n n n n n I 



30'/; 
50^4- 



■^ 



i»t>«»»r-5<>t^o;K)^!<tK;<v;^»£0!»^g<<tai>»'»»<>-v^^ 





•i.-^?5>'>?»-'^M 



Fig. 10. — Roman Corinthian. 



32 



ARCHITECTURE. 



^. M. 




41'% 



MkMUMJ^J^Uj^^M 






26^ 

.?.q;;: 







USi<!i:A\i^^<-^^,^i^^i^^^^^\s^^x\^^M^^:i^^'^ 



26^ 






■" ■■■ '" ■" ■" "' ■-' '" ■" ■■■ '" '-' "' 




iiiinnnnnnnni 

'imUMJiU 



rci 



I 



^ 



Fig. II. — CoMTOsiTE Order — Arch of Titus. 



MASSIVENESS OF EGYPTIAN STRUCTURES. 33 

mented with sculpture ; and the intercolumniation very nar- 
row, usually I J diameters and seldom exceeding 2|-. In the 
remains of a temple the walls were found to be 24 feet thick ; 
and at the gates of Thebes, the walls at the foundation were 
50 feet thick and perfectly solid. The immense stones of 
w^hich these, as well as Egyptian walls generally, were built, 
had both their inside and outside surfaces faced, and the 
oints throughout the body of the wall as perfectly close as 
upon the outer surface. For this reason, as well as that the 
buildings generally partake of the pyramidal form, arise 
their great sohdity and durability. The dimensions and ex- 
tent of the buildings may be judged from the temple of 
Jupiter at Thebes, which was 1400 feet long and 300 feet 
wide — exclusive of the porticos, of which there was a great 
number. 

It is estimated by Mr. Gliddon, U. S. Consul in Egypt, 
that not less than 25,000,000 tons of hewn stone were em- 
ployed in the erection of the Pyramids of Memphis alone — 
or enough to construct 3000 Bunker Hill monuments. Some 
of the blocks are 40 feet long, and polished with emery to a 
surprising degree. It is conjectured that the stone for these 
pyramids was brought, by rafts and canals, from a distance 
of six or seven hundred miles. 

The general appearance of the Egyptian style of archi- 
tecture is that of solemn grandeur— -amounting sometimes to 
sepulchral gloom. For this reason it is appropriate for cem- 
eteries, prisons, etc. ; and being adopted for these purposes, 
it is gradually gaining favor. 

A great dissimilarity exists in the proportion, form, and 
general features of Egyptian columns. In some instances, 
there is no uniformity even in those of the same building, 
each differing from the others either in its shaft or capital. 
For practical use in this country, Fig. 13 may be taken as a 
standard of this style. The Halls of Justice in Centre 
Street, New York Cit}^ is a building in general accordance 
with the principles of Egyptian architecture. 

4-6. — Buil€ling^*4 ill €ieiieral. — In selecting a style for an 
edifice, its peculiar requirements must be allowed to govern. 



34 



ARCHITECTURE. 




Fig, 12. — Modified Tuscan Order. 



FITNESS OF STYLES. 35 

That style of architecture is to be preferred in which utility, 
stability, and regularity are gracefully blended with gran- 
deur and elegance. But as an arrangement designed for a 
warm country would be inappropriate for a colder climate, 
it would seem that the style of building ought to be modified 
to suit the wants of the people for whom it is designed. 
High roofs to resist the pressure of heavy snows, and ar- 
rangements for artificial heat, are indispensable in northern 
climes ; while they w^ould be regarded as entirely out of 
place in buildings at the equator. 

Among the Greeks, architecture was employed chiefly 
upon their temples and other large buildings ; and the pro- 
portions of the orders, as determined by them, when execu- 
ted to such large dimensions, have the happiest effect. But 
when used for small buildings, porticos, porches, etc., espe- 
cially in country places, they are rather heavy and clumsy ; 
in such cases, more slender proportions will be found to pro- 
duce a better effect. The English cottage-style is rather 
more appropriate, and is becoming extensively practised for 
small buildings in the country. 

4-7. — ExpressioBi. — Every building should manifest its 
destination. If it be intended for national purposes, it 
should be magnificent — grand ; for a private residence, neat 
and modest ; for a banqueting-house, gay and splendid ; for 
a monument or cemetery, gloomy — melancholy ; or, if for a 
church, majestic and graceful — by some it has been said, 
'* somewhat dark and gloomy, as being favorable to a devo- 
tional state of feeling ;" but such impressions can only re- 
sult from a misapprehension of the nature of true devotion. 
"Her ways are ways oi pleasantness, and all her paths are 
peace." The church should rather be a type of that 
brighter world to which it leads. Simply for purposes of 
contemplation, however, the glare of the noonday light 
should be excluded, that the worshipper may, with Milton — 

" Love the high, embowed roof, 
With antique pillars massy proof, 
And storied windows richly dight, 
Casting a dim, religious light." 



5^ 



ARCHITECTURE. 



HP. 




Fig. 13.— Egyptian ARciiirECTURJi. 



PREVALENCE OF WOODEN DWELLINGS. 37 

However happily the several parts ot an edifice may be 
disposed, and however pleasing it may appear as a whole, 
yet much depends upon its site, as also upon the character 
and style of the structures in its immediate vicinity, and the 
degree of cultivation of the adjacent country. A splendid 
country-seat should have the out-houses and fences in the 
same style with itself, the trees and shrubbery neatly 
trimmed, and the grounds well cultivated. 

48. — Durability. — Europeans express surprise that we 
build so much with wood. And yet, in a new country, 
where wood is plenty, that this should be so is no cause for 
wonder. Still the practice should not be encouraged. Build- 
ings erected with brick or stone are far preferable to those 
of wood : the}^ are more durable ; not so liable to injury by 
lire, nor to need repairs ; and will be found in the end quite 
as economical. A wooden house is suitable for a temporary 
residence only ; and those who would bequeath a dwelling 
to their children will endeavor to build with a more dura- 
ble material. Wooden cornices and gutters, attached to 
brick houses, are objectionable^not only on account of their 
frail nature, but also because they render the building liable 
to destruction by fire. 

4-9. — ©^vciniig-HoMses : are built of various dimensions 
and styles, according to their destination ; and to give de- 
signs and directions for their erection, it is necessary to know 
their situation and object. A dwelling intended for a gar- 
dener would require very different dimensions and arrange- 
ments from one intended for a retired gentleman — with his 
servants, horses, etc. ; nor would a house designed for the 
city be appropriate for the country. For city houses, ar- 
rangements that would be convenient for one family might 
be very inconvenient for two or more. Figs. 14, 15, 16, 17, 
18, and 19 represent the icJinograpJiical projection, or ground- 
plan, of the floors of an ordinary city house, designed to be 
occupied bv one family only. Fig. 21 is an elevation, or 
front view, of the same house. All these plans are drawn at 
the same scale — which is that at the bottom of Fio: 21. 



38 ARCHITECTURE. 

Fig. 14 is a Plan of the Under-Cellar. 

a, is the coal-vault, 6 by 10 feet. 

b, is the furnace for heating the house. 

c, d, are front and rear areas. 

Fig. 15 is a Plan of the Basement. 

a, is the library, or ordinary dining-room, 15 by 20 feet. 

b, is the kitchen, 15 by 22 feet. 

c, is the store-room, 6 by 9 feet. 

d, is the pantry, 4 by 7 feet. 

c, is the china closet, 4 by 7 feet. 
/, is the servants' water-closet. 
g, is a closet. 

h, is a closet with a dumb-waiter to the first story above. 
/, is an ash closet under the front stoop, 
y, is the kitchen-range. 

k, is the sink for washing and drawing water. 
/, are wash-traj^s. 

Fig. 16 is a Plan of the First Story. 

a, is the parlor, 1 5 by 34 feet. 

b, is the dining-room, 16 by 23 feet, 
r, is the vestibule. 

Cy is the closet containing the dumb-waiter from the base- 
ment. 

f, is the closet containing butler's sink. 

g, g, are closets. 

//, is a closet for hats and cloaks. 
i,j, are front and rear balconies. 

Fig. 17 is the Second Story. 

a, a, are chambers, 15 by 13 feet. 

b, is a bed-room, 7^ by 13 feet. 

c, is the bath-room, 7 J by 13 feet. 

d, d, are dressing-rooms, 6 by 7|- feet. 
(\ c, are closets. 

/, /, are wardrobes. 
g, g, are cupboards. 



PLANS OF A CITY HOUSE. 



39 




City Dwelling. 



40 



ARCHITECTURE. 




Fig. 17. 

SECOND STORV. 



City Dwelling. 



UPPER STORIES OF A CITV HOUSE. 

Fig. 1 8 is the Third Story. 

a, a, are chambers, 15 by 19 feet. 

b, b, are bed-rooms, /i by 13 feet. 

c, r, are closets. 

d, is a linen-closet, 5 by 7 feet. 



41 




Fio. 18. 




Fi(i. 19. 


THIRD STORV. 


City Dwelling. 


POURTH STORY 



e, Cy are dressing-closets. 
/, /, are wardrobes. 
g, g, are cupboards. 

Fig. 19 is the Fourth Story. 
a, a, are chambers, 14 by 17 feet. 



42 ARCHITECTURE. 

b, b, are bed-rooms, 8^ by 17 feet. 

<r, c, c, are closets. 

d, is the step-ladder to the roof. 

Fig. 20 is the Section of the House showing the heights 
of the several stories. 

Fig. 21 is the Front Elevation. 

The size of the house is 25 feet front by 55 feet deep ; this 
is about the average depth, although some are extended to 
60 and 65 feet in depth. 

These are introduced to give some general ideas of the 
principles to be followed in designing city houses. In plac- 
ing the chimneys in the parlors, set the chimney-breasts 
equidistant from the ends of the room. The basement 
chimney-breasts may be placed nearly in the middle of the 
side of the room, as there is but one flue to pass through 
the chimney-breast above ; but in the second story, as there 
are two flues, one from the basement and one from the par- 
lor, the breast will have to be placed nearly perpendicular 
over the parlor breast, so as to receive the flues within the 
jambs of the fire-place. As it is desirable to have the chim- 
ney-breast as near the middle of the room as possible, it may 
be placed a few inches towards that point from over the 
breast below. So in arranging those of the stories above, 
always make provision for the flues from below. 

50, — Arranging the Stairs and IVindows. — There should 
be at least as much room in the passage at the side of the 
stairs as upon them ; and in regard to the length of the pas- 
sage in the second story, there must be room for the doors 
which open from each of the principal rooms into the hall, 
and more if the stairs require it. Having assigned a posi- 
tion for the stairs of the second story, now generally placed 
in the centre of the depth of the house, let the winders of 
the other stories be placed perpendicularly over and under 
them; and be careful to provide for head-room. To ascer- 
tain this, when it is doubtful, it is well to draw a vertical 
section of the whole stairs ; but in ordinary cases this is not 



FRONT OF A CITY HOUSE. 



43 



I. I 

Fig. 20. 

SECTION. 




1-^^-^^ 



to 20 



f 



30 

_J_ 



Fig. 21. 

ELEVATION. 



City Dwelling 



44 ARCHITECTURE. 

necessary. To dispose the windows properly, the middle 
window of each story should be exactly in the middle of the 
front ; but the pier between the two windows which light 
the parlor should be in the centre of that room ; because 
when chandeliers or any similar ornaments hang from the 
centre-pieces of the parlor ceilings, it is important, in order 
to give the better effect, that the pier-glasses at the front 
and rear be in a range with them. If both these objects 
cannot be attained, an approximation to each must be at- 
tempted. The piers should in no case be less in width than 
the Avindow openings, else the blinds or shutters, when 
thrown open, will interfere with one another; in general 
practice, it is well to make the outside piers f of the width 
of one of the middle piers. When this is desirable, deduct 
the amount of the three openings from the width of the 
front, and the remainder will be the amount of the width of 
all the piers ; divide this by lo, and the product will be ^ of 
a middle pier ; and then, if the parlor arrangements do not 
interfere, give twice this amount to each corner pier, and 
three times the same amount to each of the middle piers. 

51. — PriiicipBes of ArcliUccture. — To build well requires 
close attention and much experience. The science of build- 
ing is the result of centuries of study. Its progress towards 
perfection must have been exceedingly slow. In the con- 
struction of the first frail and rude habitations of men, the 
primary object was, doubtless, utility — a mere shelter from 
sun and rain. But as successive storms shattered his poor 
tenement, man was taught by experience the necessity of 
building with an idea to durability. And as the symmetry, 
proportion, and beauty of nature met his admiring gaze, 
contrasting so strangely with the misshapen and dispropor- 
tioned work of his own hands, he was led to make gradual 
changes, till his abode was rendered not only commodious 
and durable, but pleasant in its appearance ; and building 
became a line art, having utility for its basis. 

52. — ArraiigcmcKt. — In all designs for buildings of im- 
portance, utility, durability, and beauty, the first great prin- 
ciples, should be pre-eminent. In order that the edifice be 



ESSENTIAL REQUIRE.MEXTS OF A BUILDING. 45 

useful, commodious, and comfortable, the arrangement of the 
apartments should be such as to fit them for their several 
destinations ; for publio assemblies, oratory, state, visitors, 
retiring, eating, reading, sleeping, bathing, dressing, etc. — 
these should each have its own peculiar form and situation. 
To accomplish this, and at the same time to make their 
relative situation agreeable and pleasant, producing regu- 
larity and harmony, require in some instances much skill 
and sound judgment. Convenience and regularity are very 
important, and each should have due attention ; yet when 
both cannot be obtained, the latter should in most cases 
give place to the former. A building that is neither con- 
venient nor regular, wdiatever other good qualities it may 
possess, will be sure of disapprobation. 

63. — Ventilation. — Attention should be given to such 
arrangements as are calculated to promote health : among 
these, ventilation is by no means the least. For this pur- 
pose, the ceilings of the apartments should have a respect- 
able height; and the sky-light, or any part of the roof that 
can be made movable, should be arranged with cord and 
pullies, so as to be easily raised and lowered. Small open- 
ings near the ceiling, that may be closed at pleasure, should 
be made in the partitions that separate the rooms from the 
passages — especially for those rooms which are used for 
sleeping apartments. All the apartments should be so ar- 
ranged as to secure their being easily kept dry and clean. 
In dwellings, suitable apartments should be fitted up for 
batJiing with all the necessary apparatus for conveying 
Avater. 

54. — Stability. — To secure this, an edifice should be de- 
signed upon well-known geometrical principles : such as 
science has demonstrated to be necessary and sufficient for 
firmness and durability. It is well, also, that it have the 
appearance of stability as well as the reality ; for should it 
seem tottering and unsafe, the sensation of fear, rather than 
those of admiration and pleasure, Avill be excited in the be- 
holder. To secure certainty and accuracy in the applica- 
tion of those principles, a knowledge of the strength and 



46 ARCHITECTURE. 

Other properties of the materials vised is indispensable ; and 
in order that the whole design be so made as to be capable 
of execution, a practical knowledge of the requisite mechan- 
ical operations is quite important. 

65. — Decoration. — The elegance of a design, although 
chiefly depending upon a just proportion and harmony of 
the parts, will be promoted by the introduction of orna- 
ments, provided this be judiciously performed ; for enrich- 
ments should not only be ot a proper character to suit the 
style of the building, but should also have their true posi- 
tion, and be bestowed in proper quantity. The most com- 
mon fault, and one which is prominent in Roman architec- 
ture, is an excess of enrichment : an error which is carefully 
to be guarded against. But those who take the Grecian 
models for their standard will not be liable to go to that 
extreme. In ornamenting a cornice, or any other assem- 
blage ot mouldings, at least every alternate member should 
be left plain ; and those that are near the eye should be more 
finished than those which are distant. Although the charac- 
teristics of good architecture are utility and elegance, in 
connection with durability, yet some buildings are designed 
expressly for use, and others again for ornament : in the 
former, utility, and in the latter, beauty, should be the gov- 
erning principle. 

56. — Elementary Part§ of a BuiMsiig. — The builder 
should be acquainted with the principles upon which the 
essential, elementary parts of a building are founded. A 
scientific knowledge of these will insure certainty and secu- 
rity, and enable the mechanic to erect the most extensive 
and lofty edifices with confidence. The more important 
parts are the foundation, the column, the wall, the lintel, 
the arch, the vault, the dome, and the roof. A separate 
description of the peculiarities of each would seem to be 
necessary, and cannot perhaps be better expressed than in 
the following language of a modern writer on this subject, 
slightly modified : 



STRUCTURAL FEATURES OF A BUILDING. 47 

57. — The Foiiiidatioii : of a building should be begun 
at a certain depth in the earth, to secure a solid basis, below 
the reach of frost and common accidents. The most solid 
basis is rock, or gravel which has not been moved. Next 
to these are clay and sand, provided no other excavations 
have been made in the immediate neighborhood. From 
this basis a stone wall is carried up to the surface of the 
ground, and constitutes the foundation. Where it is in- 
tended that the superstructure shall press unequally, as at 
its piers, chimneys, or columns, it is sometimes of use to 
occupy the space between the points of pressure by an 
inverted arch. This distributes the pressure equally, and 
prevents the foundation from springing between the differ- 
ent points. In loose or muddy situations, it is always un- 
safe to build, unless we can reach the solid bottom below. 
In salt marshes and fiats, this is done by depositing timbers, 
or driving wooden piles into the earth, and raising walls 
upon them. The preservative quality of the salt will keep 
these timbers unimpaired for a great length of time, and 
makes the foundation equally secure with one of brick or 
stone. 

58. — The Coluinn, or Pillar: is the simplest member in 
any building, though by no means an essential one to all. 
This is a perpendicular part, commonly of equal breadth 
and thickness, not intended for the purpose of enclosure, 
but simply for the support of some part of the superstruc- 
ture. The principal force which a column has to resist is 
that of perpendicular pressure. In its shape, the shaft of a 
column should not be exactly cylindrical, but, since the 
lower part must support the weight of the superior part, in 
addition to the weight which presses equally on the whole 
column, the thickness should gradually decrease from bot- 
tom to top. The outline of columns should be a little 
curved, so as to represent a portion of a very long spheroid, 
or paraboloid, rather than of a cone. This figure is the joint 
result of two calculations, independent of beauty of appear- 
ance. One of these is, that the form best adapted for sta- 
bility of base is that of a cone ; the other is, that the figure, 



48 ARCHITECTURE. 

which would be of equal strength throughout for support- 
ing a superincumbent weight, would be generated by the 
revolution of two parabolas round the axis of the column, 
the vertices of the curves beins: at its extremities. The 
swell of the shafts of columns was called the entasis by the 
ancients. It has been lately found that the columns of the 
Parthenon, at Athens, which have been commonly supposed 
straight, deviate about an inch from a straight line, and that 
their greatest swell is at about one third of their height. 
Columns in the antique orders are usually made to diminish 
one sixth or one seventh of their diameter, and sometimes 
even one fourth. The Gothic pillar is commonly of equal 
thickness throughout. 

59. — Tlttc IVall : another elementary part of a building, 
may be considered as the lateral continuation of the column, 
answering the purpose both of enclosure and support. A 
wall must diminish as it rises, for the same reasons, and in 
the same proportion, as the column. It must diminish still 
more rapidly if it extends through several stories, support- 
ing weights at different heights. A wall, to possess the 
greatest strength, must also consist of pieces, the upper and 
lower surfaces of which are horizontal and regular, not 
rounded nor oblique. The walls of most of the ancient 
structures which have stood to the present time are con- 
structed in this manner, and frequentl}^ have their stones 
bound together with bolts and clamps of iron. The same 
method is adopted in such modern structures as are intended 
to possess great strength and durability, and, in some cases, 
the stones are even dovetailed , together, as in the light- 
houses at Eddystone and Bell Rock. But many of our 
modern stone walls, for the sake of cheapness, have only one 
face of the stones squared, the inner half of the wall being- 
completed with brick ; so that they can, in reality, be con- 
sidered only as brick walls faced with stone. Such walls are 
said to be liable to become convex outwardly, from the dif- 
ference in the shrinking of the cement. Rubble walls are 
made of rough, irregular stones, laid in mortar. The stones 
should be broken, if possible, so as to produce horizontal 



VARIOUS METHODS OF ERECTING WALLS. 49 

surfaces. The coffer walls of the ancient Romans were made 
by enclosing successive portions of the intended wall in a 
box, and filling it with stones, sand, and mortar promis- 
cuously. This kind of structure must have been extremely 
insecure. The Pantheon and various other Roman build- 
ings are surrounded with a double brick wall, having its 
vacancy filled up with loose bricks and cement. The whole 
has gradually consolidated into a mass of great firmness. 

60- — The Reticulated Walls: of the Romans — com- 
posed of bricks with oblique surfaces — would, at the present 
day, be thought highly unphilosophical. Indeed, they could 
not long have stood, had it not been for the great strength 
of their cement. Modern brick walls are laid with great 
precision, and depend for firmness more upon their position 
than upon the strength of their cement. The bricks being 
laid in horizontal courses, and continually overlaying each 
other, or breaking joints, the whole mass is strongly inter- 
woven, and bound together. Wooden walls, composed of 
timbers covered with boards, are a common but more per- 
ishable kind. They require to be constantly covered with a 
coating of a foreign substance, as paint or plaster, to pre- 
serve them from spontaneous decomposition. In some parts 
of France, and elsewhere, a kind of wall is made of earth, 
rendered compact by ramming it in moulds or cases. This 
method is called building \n pise\ and is much more durable 
than the nature of the material would lead us to suppose. 
Walls of all kinds are greatly strengthened by angles and 
curves, also by projections, such as pilasters, chimneys, and 
buttresses. These projections serve to increase the breadth 
of the foundation, and arc always to be made use of in large 
buildings, and in walls of considerable length. 

61. — TIbc IJntel, or Beam: extends in a right line over 
a vacant space, from one column or wall to another. The 
strength of the lintel will be greater in proportion as its 
transverse vertical diameter exceeds the horizontal, the 
strength being always as the square of the depth. The 
floor is the lateral continuation or connection of beams by 
means of a covering of boards. 



50 ARCHITECTURE. 

62. — The Arch : is a transverse member of a buildinof, 
answering the same purpose as the lintel, but vastly exceed- 
ing it in strength. The arch, unlike the lintel, may consist 
of D.ny number of constituent pieces, without impairing its 
strength. It is, how^ever, necessary that all the pieces should 
possess a uniform shape, — the shape of a portion of a 
wedge, — and that the joints, formed by the contact of their 
surfaces, should point towards a common centre. In this 
case, no one portion of the arch can be displaced or forced 
inward ; and the arch cannot be broken by any force which 
is not sufficient to crush the materials of which it is made. 
In arches made of common bricks, the sides of which are 
parallel, any one of the bricks might be forced inward, Avere 
it not for the adhesion of t'he cement. Any two of the bi'icks, 
however, by the disposition of their mortar, cannot collect- 
ively be forced inward. An arch of the proper form, when 
complete, is rendered stronger, instead of weaker, by the 
pressure of a considerable weight, provided this pressure be 
uniform. While building, however, it requires to be sup- 
ported by a centring of the shape of its internal surface, 
until it is complete. The upper stone of an arch is called 
the keystone, but is not more essential than any other. In 
regard to the shape of the arch, its most simple form is that 
of the semicircle. It is, however, very frequently a smaller 
arc of a circle, or a portion of an ellipse. 

63. — IIooke'§ Theory of an Arch. — The simplest theory 
of an arch supporting itself only is that of Dr. Hooke. 
The arch, when it has only its own weight to bear, may be 
considered as the inversion of a chain, suspended at each 
end. The chain hangs in such a form that the weight of 
each link or portion is held in equilibrium by the result of 
two forces acting at its extremities ; and these forces, or 
tensions, are produced, the one by the vv^eight of the portion 
of the chain below the link, the other by the same weight 
increased by that of the link itself, both of them acting ori- 
ginally in a vertical direction. Now, supposing the chain 
inverted, so as to constitute an arch of the same form and 
weight, the relative situations of the forces will be the same, 







VIADUCT AT CHAUMONT. 



PECULIARITIES OF THE ARCH. 5 I 

only they will act in contrary directions, so that they arc 
compounded in a similar manner, and balance each other on 
the same conditions. 

The arch thus formed is denominated a catenary arch. 
In common cases, it differs but little from a circular arch of 
the extent of about one third of a whole circle, and rising 
from the abutments with an obliquity of about 30 degrees 
from a perpendicular. But though the catenary arch is the 
best form for supporting its own weight, and also all addi- 
tional weight which presses in a vertical direction, it is not 
the best form to resist lateral pressure, or pressure like that 
of fluids, acting equally in all directions. Thus the arches 
of bridges and similar structures, when covered with loose 
stones and earth, are pressed sideways, as well as vertically, 
in the same manner as if they supported a weight of fluid. 
In this case, it is necessary that the arch should arise more 
perpendicularly from the abutment, and that its general fig- 
ure should be that of the longitudinal segment of an ellipse. 
In small arches, in common buildings, where the disturbing 
force is not great, it is of little consequence Avhat is the 
shape of the curve. The outlines may even be perfectly 
straight, as in the tier of bricks which we frequently sec 
over a window. This is, strictly speaking, a real arch, pro- 
vided the surfaces of the bricks tend toward a common 
centre. It is the weakest kind of arch, and a part of it 
is necessarily superfluous, since no greater portion can act 
in supportmg a weight above it than can be included be- 
tween two curved or arched lines. 

64. — Gothic Arclies. — Besides these arches, various 
others are in use. The aaite or lancet arch, much used in 
Gothic architecture, is described usually from two cen- 
tres outside the arch. It is a strong arch for supporting 
vertical pressure. The rampant arch is one in which the two 
ends spring from unequal heights. The horseshoe or Moorish 
arch is described from one or more centres placed above the 
base line. In this arch, the lower parts are in danger of 
being forced inward. The ogee arch is concavo-convex, and 
therefore fit only for ornament. 



52 ARCHITECTURE. 

65. — Arch: 9>cfinUioTis; I^inciples. — The upper sur- 
face is called the cxtrados, and the inner, the intrados. 
The spring is where the intrados meets the abutments. The 
span is the distance between the abutments. The wedge- 
shaped stones which form an arch are sometimes called 
voussoirs, the uppermost being- the keystone. The part of a 
pier from which an arch springs is called the impost, and the 
curve formed by the upper side of the voussoirs, the arcJii- 
volt. It is necessary that the walls, abutments, and piers on 
which arches are supported should be so firm as to resist the 
lateral thrust, as well as vertical pressure, of the arch. 
It will at once be seen that the lateral or sidewa}^ pressure 
of an arch is very considerable, when we recollect that every 
stone, or portion of the arch, is a wedge, a part of whose 
force acts to separate the abutments. For want of attention 
to this circumstance, important mistakes have been commit- 
ted, the strength of buildings materially impaired, and their 
ruin accelerated. In some cases, the Avant of lateral firmness 
in the walls is compensated by a bar of iron stretched across 
the span of the arch, and connecting the abutments, like the 
tie-beam of a roof. This is the case in the cathedral of Milan 
and some other Gothic buildings. 

66. — All Arcade: or continuation of arches, needs only 
that the outer supports of the terminal arches should be 
strong enough to resist horizontal pressure. In the inter- 
mediate arches, the lateral fbrce of each arch is counter- 
acted by the opposing lateral force of the one contiguous to 
it. In bridges, however, where individual arches are liable 
to be destroyed by accident, it is desirable that each of the 
piers should possess sufficient horizontal strength to resist 
the lateral pressure of the adjoining arches. 

67. — The Vault : is the lateral continuation of an arch, 
serving to cover an area or passage, and bearing the same 
relation to the arch that the wall does to the column, A 
simple vault is constructed on the principles of the arch, and 
distributes its pressure equally along the walls or abutments, 
A complex or groined \^\x\i is made by two vaults intersect- 
ing each other, in which case the pressure is thrown upon 



VARIOUS CONSTRUCTIONS OF THE DOME. 53 

springing- points, and is greatly increased at those points. 
The groined vault is common in Gothic architecture. 

68. — The Dome : sometimes called cupola, \s, a concave 
covering to a building, or part of it, and may be either a 
segment of a sphere, of a spheroid, or of any similar figure. 
When built of stone, it is a very strong kind of structure, 
even more so than the arch, since the tendency of each part 
to fall is counteracted, not only by those above and below it, 
but also by those on each side. It is only necessary that 
the constituent pieces should have a common form, and that 
this form should be somewhat like the frustum of a pyra- 
mid, so that, when placed in its situation, its four angles may 
point toward the centre, or axis, of the dome. During the 
erection of a dome, it is not necessar}^ that it should be sup- 
ported by a centring, until complete, as is done in the arch. 
Each circle of stones, when laid, is capable of supporting 
itself without aid from those above it. It follows that the 
dome may be left open at top, without a keystone, and 
yet be perfectly secure in this respect, being the reverse of 
the arch. The dome of the Pantheon, at Rome, has been 
always open at top, and 3'et has stood unimpaired for nearly 
2000 years. The upper circle of stones, though apparently 
the weakest, is nevertheless often made to support the addi- 
tional weight of a lantern or tower above it. In several of 
the largest cathedrals, there are two domes, one within the 
other, which contribute their joint support to the lantern, 
which rests upon the top. In these buildings, the dome 
rests upon a circular wall, which is supported, in its turn, by 
arches upon massive pillars or piers. This construction is 
called building upon pcndcntivcs, and gives open space and 
room for passage beneath the dome. The remarks which 
have been made in regard to the abutments of the arch 
apply equally to the walls immediately supporting a dome. 
They must be of sufficient thickness and solidity to resist 
the lateral pressure of the dome, which is very great. The 
walls of the Roman Pantheon are of great depth and solid- 
ity. In order that a dome in itself should be perfectly 
secure, its lower parts must not be- too nearly vertical, since, 



54 ARCHITECTURE. 

ill this case, they partake of the nature of perpendicular 
walls, and are acted upon by the spreading force of the 
parts above them. The dome of St. Paul's Church, in Lon- 
don, and some others of similar construction, are bound with 
chains or hoops of iron, to prevent them from spreading at 
bottom. Domes Avhich are made of wood depend, in part, 
for their strength on their internal carpentry. The Halle 
du Bled, in Paris, had originally a wooden dome more than 
200 feet in diameter, and only one foot in thickness. This 
has since been replaced by a dome of iron. (See Art- 
235.) 

69. — The Roof: is the most common and cheap method 
of covering buildings, to protect them from rain and other 
effects of the weather. It is sometimes flat, but more fre- 
quently oblique, in its shape. The flat or platform roof is 
the least advantageous for shedding rain, and is seldom used 
in northern countries. The pent roof, consisting of two 
oblique sides meeting at top, is the most common form. 
These roofs are made steepest in cold climates, where they 
are liable to be loaded with snow. Where the four sides of 
the roof are all oblique, it is denominated a hipped roof, and 
where there are two portions to the I'oof, of different ob- 
liquity, it is a curb, or mansard roof. In modern times, roofs 
are made almost exclusively of wood, though frequently 
covered with incombustible materials. The internal struc- 
ture or carpentry of roofs is a subject of considerable me- 
chanical contrivance. The roof is supported by rafters, 
w^hich abut on the walls on each side, like the extremities of 
an arch. If no other timbers existed except the rafters, 
they would exert a strong lateral pressure on the walls, 
tending to separate and overthrow them. To counteract 
this lateral force, a tie-beam, as it is called, extends across, 
receiving the ends of the rafters, and protecting the wall 
from their horizontal thrust. To prevent the tie-beam from 
sagging^ or bending downward with its own weight, a king- 
post is erected from this beam, to the upper angle of the 
rafters, serving to connect the whole, and to suspend the 
weight of the beam. This is called trussing. Queen-posts 



MANNER OF CONSTRUCTING ROOFS. 55 

are sometimes added, parallel to the king-post, in large roofs! 
also various other connecting timbers. In Gothic buildings, 
where the vaults do not admit of the use of a tie-beam, the 
rafters are prevented from spreading, as in an arch, by the 
strength, of the buttresses. 

In comparing the lateral pressure of a high roof with 
that of a low one, the length of the tie-beam being the same, 
it will be seen that a high roof, from its containing most 
materials, may produce the greatest pressure, as far as 
weight is concerned. On the other hand, if the weight of 
both be equal, then the low roof will exert the greater press- 
ure ; and this will increase in proportion to the distance of 
the point at which perpendiculars, drawn from the end of 
each rafter, w^ould meet. In roofs, as well as in wooden 
domes and bridges, the materials are subjected to an inter- 
nal strain, to resist which the cohesive strength of the ma- 
terial is relied on. On this account, beams should, when 
possible, be of one piece. Where this cannot be effected, 
two or more beams are connected together by splicing. 
Spliced beams are never so strong as whole ones, yet they 
may be made to approach the same strength, by affixing lat- 
eral pieces, or by making the ends overlay each other, and 
connecting them with bolts and straps of iron. The ten- 
dency to separate is also resisted, by letting the two pieces 
into each other by the process called scarfing. Mortices, in- 
tended to truss or suspend one piece by another, should be 
formed upon similar principles. 

Roofs in the United States, after beino^ ^boarded, receive 
a secondary covering of shingles. When intended to bo 
incombustible, they arc covered with slates or earthen tiles, 
or with sheets of lead, copper, or tinned iron. Slates are 
preferable to tiles, being lighter, and absorbing less moisture. 
Metallic sheets are chiefly used for flat roofs, wooden domes, 
and curved and angular surfaces, which require a flexible 
material to cover them, or have not a sufficient pitch to shed 
the rain from slates or shingles. Various artificial composi- 
tions are occasionally used to cover roofs, the most common 
of which are mixtures of tar with lime, and sometimes with 
sand and gravel. — Ency. Ain. (See Art. 202.) 



SECTION II.— CONSTRUCTION. 

Art. 70. — Construction X:§sentia]. — Construction is that 
part of the Science of Building which treats of the Laws of 
Pressure and the strength of materials. To the architect 
and builder a knowledge of it is absolutely essential. It de- 
serves a larger place in a volume of this kind than is gene- 
rally allotted to it. Something, indeed, has been said upon 
the styles and principles, by which the best arrangements 
may be ascertained ; yet, besides this, there is much to be 
learned. For however precise or workmanlike the several 
parts may be made, what Avill it avail, should the system of 
iraming, from deficient material, or an erroneous position of 
its timbers, fail to sustain even its own weight ? Hence the 
necessity for a knowledge of the laws of pressure and the 
strength of materials. These being once understood, Ave 
can with confidence determine the best position and dimen- 
sions for the several pieces which compose a floor or a roof, 
a partition or a bridge. As systems of framing are more or 
less exposed to heavy weights and strains, and, in case of 
failure, cause not only a loss of labor and material, but fre- 
quently that of life itself, it is very important that the mate- 
rials employed be of the proper quantity and quality to serve 
their destination. And, on the other hand, any superfluous 
material is not only useless, but a positive injury, as it is an 
unnecessary load upon the points of support. It is neces- 
sar}', therefore, to know the least quantity of material that 
will suffice for strength. Not the least common fault in 
framing is that of using an excess of material. Economy, at 
least, would seem to require that this evil be abated. 

Before proceeding to consider the principles upon which 
a system of framing should be constructed, let us attend to 
a few of the elementary laws in MccJianicSy Avhich Avill be 
found to be of great value in determining those principles. 




INTERIOR OF THE CATIIJ:DRAL. SIENNA. 



DIRI-XT AND OBLIQUE SUPPORTS. 



57 



71. — I^a\v§ of Pressure. — (i.) A heavy body always ex- 
erts a pressure equal to its own weight in a vertical direc- 
tion. Example: Suppose an iron ball weighing loo lbs. be 
supported upon the top of a perpendicular post {Fig. 22-A) ; 
then the pressure exerted upon that post will be equal to 
the weight of the ball, viz., 100 lbs. (2.) But if two inclined 
posts {Fig. 22-B) be substituted for the perpendicular sup- 
port, the united pressures upon these posts will be more 
than equal to the weight, and will be in proportion to their 
position. The farther apart their feet are spread the greater 
will be the pressure, and ince vci'sa. Hence tremendous 
strains may be exerted by a comparatively small weight. 
And it follows, therefore, that a piece of timber intended 
for a strut or post should be so placed that its axis may 
coincide, as nearly as possible, with the direction of the 
pressure. The direction of the pressure of the weight W 
{Fig. 22-B) is in the vertical line b d\ and the weight W 
would fall in that line if the two posts were removed ; hence 
the best position for a support for the weight would be in 




A. 




that line. But as it rarely occurs in systems of framing 
that weights can be supported by any single resistance, they 
requiring generally two or more supports (as in the case of 
a roof supported by its rafters), it becomes important, there- 
fore, to know the exact amount of pressure any certain 
weight is capable of exertmg upon oblique supports. Now, 
it has been ascertained that the three lines of a triangle, 
drawn parallel with the direction of three concurring forces 
in equilibrium, are in proportion respectively to these 



58 CONSTRUCTION. 

forces. For example, in Fig. 22-By we have a representation 
of three forces concurring in a point, which forces are in 
equiUbrium and at rest ; thus, the weight W is one force, 
and the resistances exerted by the two pieces of timber are 
the other two forces. The direction in which the first force 
acts is vertical — downwards ; the direction of the two other 
forces is in the axis of each piece of timber respectively. 
These three forces all tend towards the point b. 

Draw the axes a b and b c oi the two supports ; make b d 
vertical, and from d draw d e and d f parallel w4th the axes 
b c and b a repectively. Then the triangle b d c has its 
lines parallel respectively with the direction of the three 
forces ; thus, bd\s in the direction of the weight PV,dc paral- 
lel with the axis of the timber i^,and ^ ^ is in the direction of 
the timber C. In accordance with the principle above stated, 
the lengths of the sides of the triangle b d e are in propor- 
tion respectively to the three forces aforesaid ; thus — 

As the length of the line b d 

Is to the number of pounds in the weight IV, 

So is the length of the line b e 

To the number of pounds' pressure resisted by the 
timber C. 
Again — 

As the length of the line /; d 

Is to the number of pounds in the weight JF, 

So is the length of the line d c 

To the number of pounds' pressure resisted by the 
timber D. 
And again — 

As the length of the line b e 

Is to the pounds' pressure resisted by C, 

So is the length of the Ime d c 

To the pounds' pressure resisted by D. 

These proportions are more briefly stated thus — 

\st. bd\ W \\b e\ P, 

P being used as a symbol to represent the number of pounds' 
pressure resisted by the timber C. 



PARALLELOGRAM OF FORCES. 59 



O representing the number of pounds' pressure resisted by 
the timber D. 

Zd, be:P::de:Q. 

72. — Parallelog^ratn of Forces. — This relation between 
lines and pressures is applicable in ascertaining the pres- 
sures induced by known weights throughout any system of 
framing. The parallelogram b e d f is called the Parallelo- 
gram of Forces ; the two lines be and bf being called the 
components, and the line b d the resultant. Where it is re- 
quired to find the components from a given resultant [Fig. 
22-B)y the fourth line d f nt^d not be drawn, for the triangle 
b d e gives the desired result. But when the resultant is to 
be ascertained from given components {Fig. 28), it is more 
convenient to draw the fourth line. 

73b — The He§o!utBoii of Forces: is the findinsf of two 
or more forces which, acting in different directions, shall 
exactly balance the pressure of any given single force. To 
make a practical application of this, let it be required to 
ascertain the oblique pressure inFig. 22-B. In this figure the 
line bd measures half an inch (0'5 inch), and the line be 
three tenths of an inch (0-3 inch). Now if the weight W 
be supposed to be 1200 pounds, then the first stated propor- 
tion above, 

b d '. JV : : b e : P, becomes 0-5:1 200 : : o • 3 : P. 

And since the product of the means divided by one of the 
extremes gives the other extreme, this proportion n:ay be 
put in the form of an eqtiation, thus — 

1 200 X o • 3 _ p 

Performing the arithmetical operation here indicated — that 
is, multiplying together the two quantities above the line, 
and dividing the product by the quantity under the line — the 



6o 



CONSTRUCTION. 



quotient will be equal to the quantity represented by P, viz., 
the pressure resisted by the timber C. Thus — 



1 200 
0.3 

0.5)360-0 

720 = P. 

The strain upon the timber C is, therefore, equal to 720 
pounds; and since, in this case (the two timbers being- in- 
clined equally from the vertical), the line ^ ^is equal to the 
line b c, therefore the strain upon the other timber D is 
also 720 pounds. 




Fig. 23. 

74. — Iiiclinatioia of Supports Uaiequal. — In Fig. 23 the 
pressures in the two supports are unequal. The supports 
are also unequal in length. The length of the supports, 
however, does not alter the amount of pressure from the 
concentrated load supported ; but generally long timbers 
are not so capable of resistance as shorter ones. For, not 
being so stiff, they bend more readily, and, since the com- 
pression is in proportion to the length, they therefore 
shorten more. To ascertain the pressures in Fig. 23, let the 
weight suspended from b dh^ equal to two and three quarter 
tons (2-75 tons). The line b d measures five and a half 
tenths of an inch (0-55 inch), and the line b c half an inch 
(0-5 inch). Therefore, the proportion 

b d \ W w b c '. P becomes o • 5 5 : 2 • 75 : : o- 5 : P, 



JTRAIN IN PROPORTION TO INCLINATION. 6l 



and i:7Lx_o^ ^ ^, 



0-55 


2-75 
0.5 



I 10 

275 
275 



The strain upon the timber A is, therefore, equal to two 
and a half tons. 

Again, the line r d measures four tenths of an inch (0-4 
inch) ; therefore, the proportion 

b d : W :: e d : Q becomes o- 55 : 2-75 : : 0-4 : (7, 
and 





2- 


•75 X o-4_ 


-~Q: 






0.55 






2-75 








0-4 







•55) 


I • 100(2 = 
I 10 


e. 



The strain upon the timber B is. therefore, equal to two 
tons. 

75. — The Strains Exceed the ^Weights. — Thus the united 
pressures upon the two inclined supports always exceed the 
weight. In the last case, 2 J tons exert a pressure of 2^ and 
2 tons, equal together to 4I- tons ; and in the former case, 
1200 pounds exert a pressure of twice 720 pounds, equal to 
1440 pounds. The smaller the angle of inclination to the 
horizontal, the greater will be the pressure upon the sup- 
ports. So, in the frame of a roof, the strain upon the rafters 
decreases gradually with the increase of the angle of incli- 
nation to the horizon, the length of the rafter remaining the 
same. 



62 CONSTRUCTION. 

This is true in comparing one system of framing with 
another ; but in a system where the concentrated weight to 
be supported is not in the middle (see Fig. 23), and, in con- 
sequence, the supports are not inclined equally, the strain 
will be greatest upon that support which has the greatest 
inclination to the horizon. 

76. — Minimum TlBriB§t of RafteE'§. — Ordinarily, as in 
roofs, the load is not concentrated, it being that of the fram- 
ing itself. Here the amount of the load will be in proportion 
to the length of the rafter, and this will increase with the 
mcrease of the angle of inclination, the span remaining the 
same. So it is seen that in enlarging the angle of inclina- 
tion to the horizon in order to lessen the oblique thrust, the 
load is increased in consequence of the elongation of the 
rafter, thus increasing the oblique thrust. Hence there is 
an economical angle of inclination. A rafter Avill have the 
least oblique thrust when its angle of inclination to the 
horizon is 35° 16' nearly. This angle is attained very nearly 
when the rafter rises 8J inches per foot, or when the height 
B C {Fig. 32) is to the base A C as 8^ is to 12, or as 0-7071 is 
to I • o. 

77. — Practical MctSiod of Determining Strain§. — A com- 
parison of pressures in timbers, according to their position, 
may be readily made by drawing various designs of framing 
and estimating the several strains in accordance with the 
parallelogram of forces, always drawing the triangle b d e 
so that the three lines shall be parallel with the three forces 
or pressures respectively. The length of the lines forming 
this triangle is unimportant, but it will be found more con- 
venient if the line drawn parallel with the known force is 
made to contain as many inches as the known force contains 
pounds, or ai many tenths of an inch as pounds, or as many 
inches as :o is, or tenths of an inch as tons; or, in general, 
as many divisions of any convenient scale as there are units 
of weight or pressure in the known force. If drawn in this 
manner, then the number of divisions of the same scale 
found in the other two lines of the triangle will equal the 
units of pressure or weight of the other two forces respect- 



HORIZONTAL THRUST OF RAFTERS. 63 

ively, and the pressures sought will be ascertained simply by 
applying the scale to the lines of the triangle. 

For example, in Fig. 23, the vertical line b d, of the tri- 
angle, measures fifty-live hundredths of an inch (0-55 inch); 
the line be, fifty hundredths (0-50 inch); and the line e d, 
forty (0-40 inch). Now, if it be supposed that the vertical 
pressure, or the weight suspended below b d, is equal to 55 
pounds, then the pressure on A will equal 50 pounds, and 
that on B will equal 40 pounds ; for. by the proportion above 
stated, 

b d: JV:: b e : P, 
55 : 55 :: 50: 50; 

and so of the other pressure. 

If a scale cannot be had of equal proportions with the 
forces, the arithmetical process will be shortened somewhat 
by making the line of the triangle that represents the knozvn 
weight equal to unity of a decimally divided scale, then the 
other lines will be measured in tenths or hundredths; and 
in the numerical statement of the proportions between the 
lines and forces, the first term being unity, the fourth term 
will be ascertained simply by multiplying the second and 
third terms together. 

For example, if the three lines are i, 0-7, and 1-3, and 
the known weight is 6 tons, then 

b d \ IV :: b e : P becomes 
I : 6 :: o-y : P = 4'2, 

equals four and two tenths tons. Again — 

b d : W :\ c d : Q becomes 
I :6:: 1.3 : e = 7.8, 

equals seven and eight tenths tons. 

78.— Horizontal Tlirust.— In Fig. 24, the weight Impresses 
the struts in the direction of their length ; their feet, 71 11, 
therefore, tend to move in the direction no, and would so 
move Avere they not opposed by a sufficient resistance from 
the blocks, A and A. If a piece of each block be cut off at 



64 



CONSTRUCTION. 



the horizontal line, a ;/, the feet of the struts would slide 
away from each other along- that line, in the direction na; 
but if, instead of these, two pieces were cut off at the verti- 
cal line, nb, then the struts would descend vertically. To 
estimate the horizontal and the vertical pressures exerted by 
the struts, let n o be made equal (upon any scale of equal 
parts) to the number of tons with which the strut is pressed ; 




Fig. 24. 

construct the parallelogram of forces by drawing c parallel 
to an, and of parallel to bn; then nf (by the same scale) 
shows the number of tons pressure that is exerted by the 
strut in the direction n a, and ;/ e shows the amount exerted 
in the direction n b. By constructing designs similar to this, 
giving various and dissimilar positions to the struts, and 
then estimating the pressures, it will be found in every case 
that the horizontal pressure of one strut is exactly equal to 
that of the other, however much one strut may be inclined 
more than the other; and also, that the united vertical 
pressure of the two struts is exactly equal to the weight W, 
(In this calculation the weight of the timbers has not been 
taken into consideration, simply to avoid complication to 
the learner. In practice it is requisite to include the weight 
of the framing with the load upon the framing.) 

Suppose that the two struts, B and B {Fig. 24), were 
rafters of a roof, and that instead of the blocks, A and A, the 
walls of a building were the supports: then, to prevent 



TIES DESIRABLE IN ROOFS. 



65 



the walls from being thrown over by the thrust of B and B^ 
it would be desirable to remove the horizontal pressure. 
This may be done by uniting- the feet of the rafters with a 




Fig. 25. 

rope, iron rod, or piece of timber, as in Fig. 25. This figure 
is similar to the truss of a roof. The horizontal strains on 
the tie-beam, tending to pull it asunder in the direction of 
its length, may be measured at the foot of the rafter, as was 
shown at^ Fig. 24 ; but it can be more readily and as accu- 
rately measured by drawing from /and e horizontal lines to 
the vertical line, h d, meeting it in and o ; then/^ will be 
the horizontal thrust at B^ and co at A ; these will be found 
to equal one another. When the rafters of a roof are thus 
connected, all tendency to thrust out the walls horizontally 
is removed, the only pressure on them is in a vertical direc- 
tion, being equal to the weight of the roof and whatever it 
has to support. This pressure is beneficial rather than 
otherwise, as a roof having trusses thus formed, and the 
trusses well braced to each other, tends to steady the walls. 



79.— PosilioBB of Supports.— /^zVy. 26 and 27 exhibit two 
methods of supporting the equal weights, W and IV. Let 
it be required to measure and compare the strains produced 
on the pieces, A B and A C. Construct the parallelogram of 
forces, ebfd, according to Art. 71. Then b f \\\\\ show the 



66 



CONSTRUCTION. 



Strain on A B, and b e the strain on A C, By comparing the 
figures, bd being equal in each, it will be seen that the 
strains in Fig. 26 are about three times as great as those in 





Fig. 27. 



Fig. 27 ; the position of the pieces, A B and A C, in Fig 27, 
is therefore far preferable. 




Fig. 28. 



80i— The Composition of Forces : consists in ascertain- 
ing the direction and amount of one force which shall be 
just capable of balancing two or more given forces, acting in 
different directions. This is only the reverse of the resolu- 



STRAINS INVOLVED IN CRANE. 



67 



tion of forces ; and the two are founded on one and the 
same principle, and may be solved in the same manner. For 
example, let A and B {Fig. 28) be two pieces of timben 
pressed in the direction of their length towards b — A by a 
force equal to 6 tons weight, and B 9 tons. To find the 
direction and amount of pressure they would unitedly exert, 
draw the lines b e and ^/ in a line with the axes of the 
timbers, and make be equal to the pressure exerted by ^, 
viz., 9 ; also make ^/ equal to the pressure on A^ viz., 6, and 
complete the parallelogram of forces ebfd; then bd, the 
diagonal of the parallelogram, will be the direction, and its 
length, 9-25, will be the amount, of the united pressures of 
A and of B, The line b d\'s> termed the resultant of the two 
forces b f and be, U A and B are to be supported by one 
post, C, the best position for that post will be in the direc- 
tion of the diagonal bd; and it will require to be sufficiently 
strong to support the united pressures of A and of B, which 
are equal to 9-25 or 9^ tons. 




Fig. 29. 

81.— Another Example.— Let Fig. 29 represent a piece of 
framing commonly called a crane, which is used for hoist- 
ing heavy weights by means of the rope, B bf, which passes 
over a pulley at b. This, though similar to Figs. 26 and 27, is, 
however, still materially different. In those figures, the 
strain is in one direction only, viz., from b to d\ but in this 
there are two strains, from A to B and from A to IV. The 
strain in the direction A B Is evidently equal to that in the 



68 



CONSTRUCTION. 



direction A JV. To ascertain the best position for the strut 
A Cy make d e equal to hf, and complete the parallelogram of 
forces cbfd; then draw the diagonal b d, and it will be the 
position required. Should the foot, C, of the strut be placed 
either higher or lower, the strain on A C would be in- 
creased. In constructing cranes, it is advisable, in order 
that the piece B A may be under a gentle pressure, to place 
the foot of the strut a trifle lower than where the diagonal 
/;rt^ would indicate, but never higher. 



7^ 



HvJ 



Fic.. 30. 




W^vJy 



82.— Ties aii<l Struts.— Timbers in a state of tension are 
called ties, while such as are in a state of compression are 
termed stmts. This subject can be illustrated in the follow- 



Let A and B {Fig. 30) represent beams of timber sup- 
porting the weights W, IV, and JV; A having but one sup- 
port, which is in the middle of its length, and B two, one at 
each end. To show, the nature of the strains, let each beam 
be sawed in the middle from a to b. The effects are obvious : 
the cut in the beam A will open, whereas that in B will 
close. If the weights are heavy enough, the beam A will 
break at b ; while the cut in B will be closed perfectly tight 
at a, and the beam be very little injured by it. But if, on 
the other hand, the cuts be made in the bottom edge of the 
timbers, from c to b, B will be seriously injured, while A 
will scarcely be affected. By this it appears evident that, 
in apiece of timber subject to a pressure across the direction 
of its length, the fibres are exposed to contrary strains. If 
the timber is supported at both ends, as at B, those from the 
top edge down to the middle are compressed in the direction 



TIES AND STRUTS. 



69 



of their length, while those from the middle to the bottom 
edge are in a state of tension ; but if the beam is supported 
as at A, the contrary effect is produced ; while the fibres at 
the middle of either beam are not at all strained. The strains 
in a framed truss are of the same nature as those in a single 
beam. The truss for a roof, being supported at each end, 
has its tie-beam in a state of tension, while its rafters are 
compressed in the direction of their length. By this, it 
appears highly important that pieces in a state of tension 
should be distinguished from such as are compressed, in 
order that the former may be preserved continuous. A strut 
may be constructed of two or more pieces ; yet, where there 
arc many joints, it will not resist compression so well. 

83.— To Distiiig^uisb Tie§ from Struts.— This may be done 
by the following rule. In Fig: 22-B, the timbers C and JD are 
the sjistaining iorcQS, and the weight Wis the straining ioxcQ\ 
and if the support be removed, the straining force would 
move from the point of support b towards d. Let it be 
required to ascertain whether the sustaining forces arc 
stretched or pressed by the straining force. Ride : Upon the 
direction of the straining force b d^ ViS a diagonal, construct 
a parallelogram r^/^ whose sides shall be parallel with the 
direction of the sustaining forces C and D\ through the 
point b draw a line parallel to the diagonal ef] this may 
then be called the dividing line between ties and struts. 
Because all those supports which are on that side of the 
dividing line which the straining force would occupy if 
unresisted are compressed, while those on the other side of 
the dividing line are stretched. 

In Fig. 22-B, the supports are both compressed, being on 
that side of the dividing line which the straining force would 
occupy if unresisted. In Figs. 26 and 27, in which A B and 
A C are the sustaining forces, A C is compressed, whereas 
A B is in a state of tension ; A C being on that side of the 
line h /which the straining force would occupy if unresisted, 
and A B on the opposite side. The place of the latter might 
be supplied by a chain or rope. In Fig. 25, the foot of the 
rafter at A is sustained by two forces, the wall and the tie- 



70 



CONSTRUCTION. 



beam, one perpendicular and the other horizontal : the 
direction of the straining force is indicated by the line b a. 
The dividing- line h i, ascertained by the rule, shows that the 
wall is pressed and the tie-beam stretched. 




Fig, 31. 



84.— Another Example— Let ^ ^ B F (Fig. 31) represent 
a gate, supported by hinges at A and E. In this case, the 
straining force is the weight of the materials, and the direc- 
tion of course vertical. Ascertain the dividing line at the 
several points, G, B, /, J, //, and F, It will then appear that 
the force at G is sustained hj A G and G F, and the dividing 
line shows that the former is stretched and the latter com- 
pressed. The force at H is supported by u4 H and H F — the 
former stretched and the latter compressed. The force at B 
is opposed by H B and A B,one pressed, the other stretched. 
The force at F is sustained by 6^i^ and FFy GF being 
stretched and F F pressed. By this it appears that A B is in 
a state of tension, and E F oi compression ; also, that A BI 
and G F are stretched, while B H and G F are compressed : 
which shows the necessity of having^ H and 6^/^ each in 
one whole length, while B H and G F may be, as they arc 
shown, each in two pieces. The force at y is sustained by 
GJ and J H, the former stretched and the latter compressed. 
The piece C D is neither stretched nor pressed, and could 
be dispensed Avith if the joinings at y and /could be made 



TO FIND THE CENTRE OF GRAVITY. 71 

as effectually without it. In case A B should fail, then C 1) 
would be in a state of tension. 

86. — Centre of Gravity. — The centre of gravity of a uni- 
form prism or cylinder is in its axis, at the middle of its 
length ; that of a triangle is in a line drawn from one angle 
to the middle of the opposite side, and at one third of the 
length of the line from that side ; that of a right-angled tri- 
angle, at a point distant from the perpendicular equal to one 
third of the base, and distant from the base equal to one 
third of the perpendicular ; that of a pyramid or cone, in 
the axis and at one quarter of the height from the base. 

The centre of gravity of a trapezoid (a four-sided figure 
having only two of its sides parallel) is in a line joining the 
centres of the two parallel sides, and at a distance from the 
longest of the parallel sides equal to the product of the 
length in the sum of twice the shorter added to the longer 
of the parallel sides, divided by three times the sum of the 
two parallel sides. Algebraically thus : 

/ (2 ^ + Z') 

where d equals the distance from the longest of the parallel 
sides, / the length of the line joining the two parallel sides, 
and a the shorter and b the longer of the parallel sides. 

Example, — A rafter 25 feet long has the larger end 14 
inches wide, and the smaller end 10 inches wide: how far 
from the larger end is the centre of gravity located ? 

Here I z=z 2^, a = -}-f , and b — i|, 

hence d = ^(^^ + ^^) =. -5(2 X If + 11-) ^ 25^x_-||. ^ 25^x34 ^ 
3{^ + /0 3 (if + if) 3xft 3x24 

-^^^- = 11-8 = II feet Q# inches nearlv. 

In irregular bodies with plain sides, the centre of gravity 
may be found by balancing them upon the edge of a prism 
— upon the edge of a table — in two positions, making a line 
each time upon the body in a line with the edge of the prism, 
and the intersection of those lines will indicate the point re- 



72 



CONSTRUCTION. 



quired. Or suspend the article by a cord or tliread attached 
to one corner or edge ; also from the same point of suspen- 
sion hang a plumb-line, and mark i(s position on the face of 
the article; again, suspend the article from another corner 
or side (nearly at right angles to its former position), and 
mark the position oi the plumb-line upon its face ; then the 
intersection of the two lines will be the centre of gravity. 




Fin. 32. 

86. — Effect of the Weight of Inoliiie<1 Beani§. — An in- 
clined post or strut supporting some heavy pressure applied 
at its upper end, as at Fi^. 25, exerts a pressure at its foot in 
the direction of its length, or nearly so. But when such a 
beam is loaded uniformly over its whole length, as the rafter 
of a roof, the pressure at its foot varies considerably from 
the direction of its length. For example, let A B {Fig- 32) 
be a beam leaning against the wall B r, and supported at its 
loot by the abutment A, in the beam A c, and let o be the 
centre of gravity of the beam. Through o draw the verti- 
cal line b d, and from B draw the horizontal line B b, cutting 
b dm b\ join b and A, and b A will be the direction of the 
thrust. To prevent the beam from loosing its footing, the 
joint at A should be made at right angles to, bA. The 
amount of pressure will be found thus : Let b d (by any scale 
of equal parts) equal the number of tons upon the beam 
A B ; draw d e parallel to B b ; then /; c (by the same scale) 
equals the pressure in the direction /; A ; and c d the pres- 
sure against the w^all at B — and also the horizontal thrust ac 
A^ as these are always equal in a construction of this kind. 

The horizontal thrust of an inclined beam {Fig. 32) — the 
effect of its own Aveight — may be calculated thus: 

Rule. — Multiply the weight of the beam in pounds by 



THRUST OF INXJTXED DKAMS. 



75 



its base, A C, in feet, and by the distance in feet of its centre 
of gravity, o (see Art. 85), from tiie lower end, at A, and 
divide this product by the product of the length, A B, into 
the height, B C, and the quotient will be the horizontal 

thrust in pounds. This may be stated thus: H =1 —-A 

where d equals the distance of the centre of gravity, 0, from 
the lower end ; b equals the base, A C ; iv equals the weight 
of the beam ; h equals the height, B C ; /equals the length 
of the beam; and //equals the horizontal thrust. 

Example. — Abeam 20 feet long weighs 300 pounds; its 
centre of gravity is at 9 feet from its lower end ; it is so in- 
clined that its base is 16 feet and its height 12 feet : what is 
the horizontal thrust ? 

TT d b w, 0x16x300 0x4x21; 

Here — — — becomes — — = - — ~ = x 4 x s 

hi 12x20 5 J -t J 

= 180 =1/ — the horizontal thrust. 

This rule is for cases where the centre of gravity does 
not occur at the middle of the length of the beam, although 
it is applicable when it docs occur at the middle ; yet a 
shorter rule Avill sufBce in this case, and it is thus: 

Ride. — Multiply the weight of the rafter in pounds by 
the base, A C {Fig. 32), in feet, and divide the product by 
twice the height, B C, in feet, and the quotient will be the 
horizontal thrust, when the centre of gravit}' occurs at the 
middle of the beam. 

If the inclined beam is loaded with an equally distributed 
load, add this load to the weight of the beam, and use this 
total weight in the rule instead of the weight of the beam. 
And generally, if the centre of gravity of the combined 
weights of the beam and load does not occur at the centre 
of the length of the beam, then the former rule is to be used. 

In Fig. 33, two equal beams are supported at their feet by 
the abutments in the tie-beam. This case is similar to the 
last; for it is obvious that each beam is in precisely the 
position of the beam in Fig. 32. The horizontal pressures at 
B, being equal and opposite, balance one another; and their 
horizontal thrusts at the tie-beam are also equal. (See Art. 



74 



CONSTRUCTION. 



78 — Fig. 25.) When the height of a roof {Fig. 33) is one 
fourth of the span, or of a shed {Fig. 32) is one halt the 
span, the horizontal thrust of a rafter, whose centre of grav- 




FiG. 33. 

ity is at the middle of its length, is exactly equal to the 
weight distributed uniformly over its surface. 

In shed or lean-to roofs, as Fig. 32, the horizontal pressure 
will be entirely remov^ed if the bearings of the rafters, as A 
and B {Fig. 34), are made horizontal — provided, however, 




^^. 



Fig. 34. 



that the rafters and other framing do not bend between the 
points of support. If a beam or rafter have a natural curve, 
the convex or rounding edge should be laid uppermost. 

87. — Effect of Load on Beam. — The strain in a uniformly 
loaded beam, supported at each end, is greatest at the 
middle of its length. Hence mortices, large knots, and other 
defects should be kept as far as possible from that point ; 
and in resting a load upon a beam, as a partition upon a 
floor-beam, the weight should be so adjusted, if possible, 
that it will bear at or near the ends. 

Twice the weight that will break a beam, acting at the 
centre of its length, is required to break it wdien equally 



VARYING PRESSURE ON HEARINGS. 



/O 



distributed over its length ; and precisely the same deflec- 
tion or sdor will be produced on a beam by a load equally 
distributed that five eighths ot the load will produce if act- 



ing at the centre of its length. 



88. — Eifert on Bearings. — When a uniformly loaded 
beam is supported at each end on level bearings (the beam 
itself being either horizontal or inclined), the amount of 
pressure caused by the load on each point of support is 
equal to one half the load ; and this is also the ase when 
the load is concentrated at the middle of the beam, or has 
its centre of gravity at the middle of the beam ; but when 
the load is unequally distributed, or concentrated so that its 
centre of gravity occurs at some other point than the middle 
of the beam, then the amount of pressure caused by the 
load on one of the points of support is unequal to that on 
the other. The precise amount on each may be ascertained 
by the following rule. 

Ru/e. — Multiply the weight JV {Fig. 35) by its distance, 
C B, from its nearest point of support, ^, and divide the pro- 
duct by the length, AB, of the beam, and the quotient will 




A 



Fig 35. 

be the amount of pressure on the rcinoie point of support. A, 
Again, deduct this amount from the weight W, and the re- 
mainder will be the amount of pressure on the near point of 
support, B\ or, multiply the weight W by its distance, A C, 
from the remote point of support. A, and divide the pro- 
duct by the length, A B, and the quotient will be the amount 
of pressure on the near point of support, B. 

When / equals the length between the bearings A and B, 
n ~ A C, in — C B, and W = the load ; then 



76 CONSTRUCTION. 

— - — = A — the amount of pressure at A, and 

-— — z= B := the amount of pressure at B. 

Example. — A beam 20 feet long- between the bearings 
has a load of 100 pounds concentrated at 3 feet from one of 
the bearings: what is the portion of this weight sustained by 
each bearing? 

Here JF= 100; n, 17; ;;/, 3; and /, 20. 

n A Win 100x3 

Hence A = — -— = = 15. 

/ 20 

Load on A = 15 pounds. 
Load on B =85 pounds. 
Total weight = 100 pounds. 

RESISTANCE OF MATERIALS. 

89. — Weiglit— Strength. — Preliminary to designing a roof- 
truss or other piece of framing, a knowledge of two subjects 
is essential : one is, the effect of gravity acting upon the 
various parts of the intended structure ; the other, the powxr 
of resistance possessed by the materials of which the framing 
is to be constructed. The former subject having been 
treated of in the preceding pages, it remains now to call at- 
tention to the latter. 

90. — <^uality of Materials. — Materials used in construc- 
tion are constituted in their structure either of fibres 
(threads) or of grains, and are termed, the former fibrous, 
the latter granular. All woods and wrought metals are 
fibrous, while cast iron, stone, glass, etc., are granular. The 
strength of a granular material lies in the power of attrac- 
tion acting among the grains of matter of which the mate- 
rial is composed, by which it resists any attempt to separate 
its grams or particles of matter. A fibre of wood or of 



THE THREE KINDS OP^ STRAINS. 



77 



wrought metal has a strength by which it resists being com- 
pressed or shortened, and finally crushed ; also a strength 
by which it resists being extended or made longer, and 
hnally sundered. There is another kind of strength in a 
fibrous material : it is the adhesion of one fibre to another 
along their sides, or the lateral adhesion of the fibres. 

91. — Iflaiiiicr of Rcsi»tiii§^. — In the strain applied to a 
post supporting a weight imposed upon it (^Fig, 36), we 
have an instance of an essay to shorten the fibres of w^hich 
the timber is composed. The strength of the timber in 
this case is termed the resistance to compression. In the strain 
on a piece of timber like a king-post or suspending piece 
{A, Fig. 37), we have an instance of an essay to extend or 
lengthen the fibres of the material. The strength here ex- 
hibited is termed the resistance io tension. When a piece of 
timber is strained like a floor-beam or any horizontal piece 




Fig. 36. 





Fig 38. 



I 



carrying a load {Fig. 38). we have an instance in which the 
two strains of compression and tension are both brought 
mto action ; the fibres of the upper portion of the beam be- 
ing compressed, and those of the under part being stretched. 



78 CONSTRUCTION. 

This kind of strength of timber is termed resistance to cross- 
strains. In each of these three kinds of strain to Avhich tim- 
ber is subjected, the power of resistance is in a measure due 
to the lateral adhesion of the fibres, not so much perhaps in 
the simple tensile strain, yet to a considerable degree in the 
compressive and cross strains. But the power of timber, 
by which it resists a pressure acting compressively in the 
direction of the length of the fibres, tending to separate the 
timber by splitting off a part, as in the case of the end of a 
tie-beam, against which the foot of the rafter presses, is 
wholly due to the lateral adhesion of the fibres. 

92. — Strength and Stiffness. — The strength of materials 
is their power to resist fracture, while the stiffness of mate- 
rials is their capability to resist deflection or sagging. A 
knowledge of their strength is useful, in order to determine 
their limits of size to sustain given weights safely ; but a 
knowledge of their stiffness is more important, as in almost 
all constructions it is desirable not only that the load be safely 
sustained, but that no appearance of v/eakness be manifested 
by any sensible deflection or sagging. 

93. — Experiments : Constants. — In the investigation of 
the laws applicable to the resistance of materials, it is found 
that the dimensions — length, breadth, and thickness — bear 
certain relations to the weight or pressure to which the 
piece is subjected. These relations are general ; they exist 
quite independently of the peculiarities of any specific piece 
of material. These proportions between the dimensions 
and the load are found to exist alike in wood, metal, stone, 
and glass, or other material. One law applies alike to all 
materials ; but the capability of materials to resist differs in 
accordance with the compactness and cohesion of particles, 
and the tenacity and adhesion of fibres, those qualities upon 
which depends the superiority of one kind of material over 
another. The capability of each particular kind of material 
is ascertained by experiments, made upon several specimens, 
and an average of the results thus obtained is taken as an 
index of the capability of that material, and is introduced 
in the rules as a constant number, each specific kind of ma- 



VALUES OF WOODS FOR COMPRKSSION. 79 

terial having its own special constant^ obtained by ex- 
perimenting- on specimens of that peculiar material. The 
results of experiments made to test the resistance of various 
materials useful in construction — their capability to resist 
the three strains before named — will now be introduced. 

94. — Resistance to Compression. — The following table 
exhibits the results of experiments made to test the resist- 
ance to compression of such woods as are in common use in 
this country for the purposes of construction. 

Table I. — Resistance to Compression. 



Material. 





C. 


//. 


io. 












m 




£-5 







o.S 


^.S 




X.'O 


c.'H 


^ 10 


tC 


s -• 


K ^ 


.n"<' 


I 


v^ 


t^ 


- >> 

CIS 


m 


^- 





S 






H 


H > 




Pounds 


Pounds 


Pounds 




per inch. 


per inch. 


per inch. 


0-613 


9500 


840 


2250 


0-762 


1 1 700 


1 160 


2800 


0-774 


8000 


1250 


2650 


0-369 


7S50 


540 


650 


0-388 


6650 


480 


800 


0-423 


5700 


370 


800 


0-397 


3400 




800 


0-491 


6700 




1250 


0-517 


5850 


.... 


3100 


0-574 


8450 




2700 


0-877 


13750 




4100 


0-494 


9050 




2500 


0-421 


7800 




2100 


0-837 


1 1 600 


.... 


5700 


0-439 


4900 


.... 


1700 


0-916 


moo 




6S0O 


1-282 


1 2 100 


.... 


7700 



r. 



Georgia Pine 

Locust 

White Oak 

Spruce 

White Pine 

Hemlock 

White Wood 

Chestnut 

Ash 

Maple 

Hickory 

Cherry 

Black Walnut 

Mahogany (St. Domingo) 
(Bay Wood) .. 

Live Oak 

Lignum Vitae 



900 
1 1 20 

1060 

260 

320 

320 

320 

500 

1240 

1080 

1640 

1000 

840 

2280 

680 

2720 

3080 



The resistance of timber of the same name varies much ; 
depending as it obviously must on the soil in which it grew, 
on its age before and after cutting, on the time of year 
when cut, and on the manner in which it has been kept since 
it was cut. And of wood from the same tree much depends 
upon its location, whether at the butt or towards the limbs, 
and whether at the heart or of the sap, or at a point mid- 
way from the centre to the circumference of the tree. The 



80 CONSTRUCTION. 

pieces submitted to experiment were of ordinary good 
quality, such as would be deemed proper to be used in 
framing. The prisms crushed were generally small, about 
2 inches long, and from i inch to i^ inches square ; some 
were Avider one way than the other, but all containing in 
area of cross section from i to 2 inches. The weight given 
in the table is the average weight per superficial inch. 

Of the first six woods named, there were nine specimens 
of each tested ; of the others, generally three specimens. 

The results for the first six woods named are taken 
from the author's work on Transverse Strams, published by 
John Wiley & Sons, New York. The results for these six 
woods, as well as those for all the others named in the table, 
were obtained by experiments carefully made by the author. 
The first six woods named Avere tested in 1874 and 1876, and 
upon a testing machine, in which the power is transmitted 
to the pieces tested, by levers acting upon knife-edges. 
For a description of this machine, see Transverse Strains, 
Art. 704. The woods named in the table, other than the 
first six, Avere tested some twenty years since, and upon a 
hydraulic press, which, owing to friction, gave results too 
low. 

The results, as thus ascertained, were given to the public 
in the 7th edition of this work, in 1857. I^^ the present edi- 
tion, the figures in Table L, for these woods, are those 
Avhich have resulted by adding to the results given by the 
hydraulic press a certain quantity thought to be requisite 
to compensate for the loss by friction. Thus corrected, the 
figures in the table may be taken as sufficiently near approx- 
imations for use in the rules, — although not so trustworthy 
as the results given for the first six woods named, as these 
were obtained upon a superior testing machine, as above 
stated. 

In the preceding table, the second column contains the 
specific gravity of the several kinds of wood, showing their 
comparative density. The weight in pounds of a cubic foot 
of any kind of wood or other material is equal to its 
specific gravity multiplied by 62-5, this number being the 
weight in pounds of a cubic foot of water. The third column 



EXPLANATION OF TABLE L 8 1 

contains the weight in pounds required to crush a prism 
having a base of one inch square ; the pressure applied to 
the fibres longitudinally. In practice, it is usual never to 
load material exposed to compression with more than one 
fourth of the crushing weight, and generally with from one 
sixth to one tenth only. The fourth column contains the 
weight in pounds which, applied in line with the length of the 
fibres, is required to force off a part of the piece, causing the 
iibres to separate by sliding, the surface separated being one 
inch square. The fifth column contains the weight in pounds 
required to crush the piece when the pressure is applied to the 
fibres transversely, the piece being one inch thick, and the 
surface crushed being one inch square, and depressed one 
twentieth of an inch deep. The sixth column contains the 
value of P in the rules; P being the weight in pounds, ap- 
plied to the iibres transversely, which is required to make a 
sensible impression one inch square on the side of the piece, 
this being the greatest weight that would be proper for a 
post to be loaded with per inch surface of bearing, resting 
on the side of the kind of wood set opposite in the table. A 
greater weight would, in proportion to the excess, crush the 
side of the wood under the post, and proportionably derange 
the framing, if not cause a total failure. It will be observed 
that the measure of this resistance is useful in limiting the 
load on a post according to the kind of material contained, 
not in the/(?5/, but in the timber upon which the post presses. 

95. — Rcsistaatcc to Tension. — The resistance of materials 
to the force of stretching, as exemplified in the case of a 
rope from which a weight is suspended, is termed the resist- 
ance to tension. In fibrous materials, this force will be differ- 
ent in the same specimen, in accordance with the direction 
in which the force acts, whether in the direction of the length 
of the fibres or at right angles to the direction of their length. 
It has been found that, in hard Avoods, the resistance in the 
former direction is about eisfht to ten times what it is in the 
latter ; and in soft woods, straight grained, such as white 
pine, the resistance is from sixteen to twenty times. A 
knowledge of the resistance in the direction of the fibres is 
the most useful in practice. 



82 



CONSTRUCTION. 



In the following table arc recorded the results of ex- 
periments made to test this resistance in some of the woods 
in common use, and also in iron, cast and wrought. Each 
specimen of the woods was turned cylindrical, and about 2 
inches diameter, and then the middle part reduced to about 
f of an inch diameter, at the middle of the reduced part, 
and thence gradually increased toward each end, where it 
was considerably larger at its junction with the enlarged 
end. The results, in the case of the iron and of the first six 
woods named, are taken from the author's work, Trans- 
verse Strains, Table XX. Experiments were made upon 
the other three woods named by a hydraulic press, some 
twenty years since, and the results were first published in 
the 7th edition of this work, in 1857. These results, owing 
to friction, wxre too low. Adding to them what is supposed 
to be the loss by friction of the machine, the results thus 
corrected are what are given for these three woods in the 
following table, and may be taken as fair approximations, 
but are not so trustworthy as the figures given for the other 
six woods and for the metals. 

Table II. — Resistanxe to Tension. 



Material. 



Georgia Pine 

Locust 

White Oak 

Spruce 

White Pine 

Hemlock 

Hickory. . . 

Maple 

Ash 

Cast Iron, American ) from 

English \ to 

Wrought Iron, American \ from 

" English \ to 



Specific 
Gravity. 



0-65 
0.794 

o 774 
o 432 

0-458 

0402 

0-75I 
0694 
o-6oS 
6-944 

7 584 
7 600 
7-792 



Pounds re- 
quired to rup- 
ture one inch 
bquare. 



16OCO 
24800 
I95CO 
19500 
I20C0 
8700 
26OCO 
20000 
15000 
27000 
17000 

Coooo 
50COO 



The figures in the table denote the ultimate capability of 
a bar one inch square, or the weight in pounds required to 



VALUES C)i'^ MATERIALS FOR CROSS-STRAINS. 



83 



produce rupture. Just what portion of this should be taken 
as the safe capabiHty will depend upon the nature of the 
strain to which the material is to be exposed. In practice it 
is found that, through defects in workmanship, the attach- 
ments may be so made as to cause the strain to act along one 
side of the piece, instead of through its axis; and that in this 
case fracture will be produced with one third of the strain 
that can be sustained through the axis. Due to this and 
other contingencies, it is usual to subject materials exposed 
to tensile strain with only from one sixth to one tenth of 
the breaking weight. 

96, — Resistance to Transverse Strains. — In the follow- 
ing table are recorded the results of experiments made to 
test the capability of the various materials named to resist 
the effects of transverse strain. The figures are taken from 
the author's work. Transverse Strains^ before referred to. 

Table III. — Transverse Strains. 



Material. 



Georgia Pine 

Locust 

White Oak . 

Spruce 

White Pine 

Hemlock 

White Wood 

Chestnut 

Ash 

Maple 

Hickory 

Cherry 

Black Walnut 

Mahogany (St. Domingo) 

(Bay Wood).. 

Cast Iron, American .... 

" English 

Wrought Iron, American 
" English . . 

Steel, in Bars 

Blue Stone Flagging . . . 

Sand Stone 

Brick, common 

" pressed 

Marble, East Chester . . . 



B. 

Resistance 

to 
Rupture. 



850 

1200 

650 

500 

450 
600 
480 
900 
1 100 
1050 
650 

650 
850 
2500 
2100 
2600 
1900 
6000 
200 

59 

33 

37 

147 



F. 

Resistance 

to 
Flexure. 



5900 

5050 

3100 

3500 

2900 

2800 

3450 

2550 

4000 

5150 

3850 

2850 

- 3900 

3600 

4750 

50000 

40000 

62000 

60000 

70000 



Extension 

of 

Fibres. 



•00109 

•0015 

•00086 

• 00098 
■0014 
•00095 

• 00096 
•00103 
■001 I I 

0014 
•0013 
•001563 
•00104 
■OOII6 

00109 



Margin 

for 
Safety. 



1-84 
2 -20 

3-39 
2-23 

I-7I 

35 
52 
54 
82 
12 
91 
03 
57 
16 
28 



84 CONSTRUCTION. 

The figures in the second column, headed B, denote the 
weight in pounds required to break a luiit of the material 
named when suspended from the middle, the piece being 
supported at each end. The unit of material is a bar one inch 
square and one foot long between the bearings. The third 
column, headed F, contains the values of the several mate- 
rials named as to their resistance to flexure, as explained in 
Arts. 302-305, Transverse Strains. These values of F, as 
constants, are used in the rules. The fourth column, headed 
Cy contains the values of the several materials named, denot- 
ing the elasticity of the fibres, as explained in Art. 312, 
Transverse Strains. These values of r, as constants, are 
used in the rules. 

The fifth column, headed a, contains for the several ma- 
terials named the ratio of the resistance to flexure as com- 
pared with that to rupture, and which, as constants used 
in the rules, indicate the margin of safety to be given for 
each kind of material. The figures given in each case show 
the smallest possible A'alue that may be safely given to a, the 
factor of safety. In practice it is generally taken higher 
than the amount given in the table. For example, in the 
table the value of B, the constant for rupture b}^ transverse 
strain for spruce, is 550. 

Now, if the dimensions of a spruce beam to carry a given 
weight be computed by the rules, using the constant B, at 
550, the beam will be of such a size that the given weight 
will just break it. 

But if, in the computation, instead of taking the full 
value of B, only a part of it be taken, then the beam will not 
break immediately ; and if the part taken be so small that 
its effect upon the fibres shall not be sufficient to strain them 
beyond their limit of elasticity, the beam will be capable of 
sustaining the weight for an indefinite period ; in this case 
the beam will be loaded by what is termed the safe weight. 
Or, since the value oi a for spruce is 2-23 in the table, if, in- 
stead of taking B at 550, its full value, only the quotient 
arising from a division of B by a be taken — or 550 divided 
by 2-23, which equals 246-6 — then the beam will be of suffi- 
cient size to carry the load safely. Therefore, while the con- 



THE VAKIOUS CLASSES OF PRESSURES. 85 

stant B is to be used for a breaking- weight, for a safe load 

P 
the quotient of — is to be used* But, again, if a be taken at 

a 

its value as given in the table, the computed beam v/ili be 
loaded up to its limit of safety. So loaded that, if the load 
be increased only in a small degree, the limit of safety will 
be passed, and the beam liable, in time, to fail by rupture. 

Therefore, as the values of a, in the table, are the smallest 
possible, it is prudent in practice always to take a larger 
than the table value. For example, the table value of 
a for spruce is 2-23, but in practice let it be taken at 3 
or 4. 



97. — Resistance to Compression. — The resistance of ma- 
terials to the force of compression may be considered in 
several ways. Posts having their heights less than ten 
times their least sides will crush before bending ; these 
belong- to one class : another class is that which com- 
prises all posts the height of which is equal to ten times 
their least sides, or more than ten times ; these will bend 
before crushing. Then there remains to be considered 
the manner in which the pressure is applied : whether in 
line with the fibres, or transversely to them ; and, again, 
whether the pressure tends to crush the fibres, or simply 
to force off a part of the piece by splitting. The various 
pressures may be comprised in the four classes following, 
namely: 

1st. — When the pressure is applied to the fibres trans- 
versely. 

2d.— When the pressure is applied to the fibres longi- 
tudinally, and so as to split off the part pressed against, 
causing the fibres to separate by sliding. 

3d. — When the pressure is applied to the fibres longi- 
tudinally, and on short pieces. 

4th. — When the pressure is applied to the fibres longi- 
tudinall}^, and on long pieces. 

These four classes will now be considered in their reg- 
ular order. 



86 CONSTRUCTION. 

98. — Compression Transversely to the Fibres. — In this 
first class of compression, experiment has shown that the 
resistance is in proportion to the number of fibres pressed, 
that is, in proportion to the area. For example, if 5000 
pounds is required to crush a prism with a base i inch 
square, it will require 20,000 pounds to crush a prism having 
a base of 2 by 2 inches, equal to 4 inches area ; because 4 
times 5000 equals 20000. 

Therefore, if any given surface pressed be multiplied by 
the pressure per inch which the kind of material pressed 
ma}^ be safely trusted with, the product will be the total 
pressure which may with safety be put upon the given sur- 
face. Now, the capability for this kind of resistance is given 
in column P, in Table I., for each kind of material named in 
the table. Therefore, to find the limit of weight, proceed 
as follows: 

99. — The liiinit of IVeiglit. — To ascertain what weight 
a post may be loaded with, so as not to crush the surface 
against which it presses, we have — 

Rule 1. — Multiply the area of the post in inches by the 
value of P, Table I., and the product will be the weight re- 
quired in pounds ; or — 

W^AP. (i.) 

Example, — A post, 8 by 10 inches, stands upon a white- 
pine girder; the area equals 8 x 10 = 80 inches. This being 
multiplied by 320, the value of /^, Table I., set opposite white 
pine, the product, 25600, is the required weight in pounds. 

100. — Area of Post.— To ascertain what area a post must 
have in order to prevent the post, loaded with a given 
weight, from crushing the surface against which it presses, 
we have — 

Rule II. — Divide the given weight in pounds by the value 
of P, Table I., and the quotient will be the area required in 
inches ; or — 



RESISTANCE TO RUPTURE UV SLIDING. 8/ 

Example. — A post standing on a Georgia-pine girder is 
loaded with 100,000 pounds: what must be its area? The 
weight, 1 00000, divided by 900, the value of P, Table I., set 
opposite Georgia pine, the quotient, in • 11, is the required 
area in inches. The post may be 10 by 11-^, or 10 by 11 
mches; or if square each side will be 10-54 inches, or w^^ 
inches diameter if round. 

101.— Riaptiirc by Sliding. — In this the second class of 
rupture by compression, it has been ascertained by ex- 
periment that the resistance is in proportion to the area of 
the surface separated without regard to the form of the sur- 
face. Now, in Table I., column //, we have the ultimate 
resistance to this strain of the several materials named. 
But to obtain the safe load per inch, the ultimate resist- 
ance of the table is to be divided by a factor of safety, of 
such value as circumstances may seem to require. Gener- 
ally this factor may be taken at 3. Then to obtain the safe, 
load for any given case, we have but to multiply the given 
surface by the ultimate resistance, and divide by the factor 
of safety ; therefore, proceed as follows : 

102- — TJae Limit of WclgliJ. — To ascertain what weight 
may be sustained safely by the resistance of a given area of 
surface, when the weight tends to split off the part pressed 
against by causing, in case of fracture, one surface to slide 
on the other, we have — 

Rule III. — Multiply the area of the surface by the value 
of H, in Table I. divide by the factor of safety, and the 
quotient w^ill be the weight required in pounds ; or — 

IF = 4^ (3.) 

a 

Example. — The foot of a rafter is framed into the end of 
its tie-beam, so that the uncut substance of the tie-beam is 
I 5 inches long from the end of the tie-beam to the joint of 
the rafter ; the tie-beam is of white pine, and is 6 inches 
thick: what amount of horizontal thrust will this end of 
the tie-beam sustain, without danger of having the end o| 



88 CONSTRUCTION. 

the tie-beam split off? Here the area of surface that sus- 
tains the pressure is 6 by 15 inches, equal to 90 inches. 
This multiplied by 480, the value of //, set opposite to 
white pine, Table I., and divided by 3, as a factor of safety, 
gives a quotient of 14400, and this is the required weight in 
pounds. 

103- — Area of Surface— To ascertain the area of surface 
that is required to sustain a given Aveight safely, when the 
weight tends to split off the part pressed against, by causing, 
in case of fracture, one surface to slide on the other, we 
have — 

Rule \Y . — Divide the given weight in pounds by the 
value of Hy Table I. ; multiply the quotient by the factor of 
safety, and the product will be the Required area in inches ; 
or — 

^ = -77- ^-^ 

Example. — The load on a rafter causes a horizontal thrust 
at its foot of 40,000 pounds, tending to split off the end of 
the tie-beam : what must be the length of the tie-beam be- 
yond the line where the foot of the rafter is framed into it, 
the tie-beam being of Georgia pine, and 9 inches thick ? 
The weight, or horizontal thrust, 40000, divided by 840, the 
value of //, Table I., set opposite Georgia pine, gives a quo- 
tient of 47-619, and this multiplied by 3, as a factor of safety, 
gives a product of 142-857. This, the area of surface in 
inches, divided by 9, the breadth of the surface strained 
(equal to the thickness of the tie-beam), the quotient, 15.87, 
is the length in inches from the end of the tie-beam to the 
rafter joint, say 16 inches. 

104.— Tenons and Splices.— A knowledge of this kind of 
resistance of materials is useful, also, in ascertaining the 
length of framed tenons, so as to prevent the pin, or key, 
with which they are fastened from tearing out ; and, also, in 
cases where tie-beams, or other timber under a tensile strain, 



CRUSiriNG STRENGTH OF STOUT POSTS. 89 

are spliced, this rule gives the length of the jog-gle at each 
end of the splice. 

105. — §toui P©§ts.— These comprise the third class of ob- 
jects subject to compression (Art. 97), and include all posts 
which are less than ten diameters high. The resistance to 
compression, in this class, is ascertained to be directly in pro- 
portion to the area of cross-section of the post. 

Now in Table I., column C, the ultimate resistance to 
crushing is given for the several kinds of materials named ; 
from Avhich the safe resistance per inch may be obtained by 
dividing it by a proper factor of safety. Having the safe 
resistance per inch, the resistance of any given post may be 
determined b}^ multiplying it by the area of the cross-section 
of the post. Therefore, proceed as follows : 

106. — TBie L.Qinit of WefgBU. — To find the weight that can 
be safely sustained by a post, when the height of the post is 
less than ten times the diameter if round, or ten times the 
thickness if rectangular, and the direction of the pressure 
coinciding with the axis, we have — 

Rule V. — Multiply the area of the cross-section of the 
post in inches by the value of C, in Table I. ; divide the pro- 
duct by the factor of safety, and the quotient will be the re- 
quired weight in pounds ; or — 

W^ ™- ■ (5-) 



Example. — A Georgia-pine post is 6 feet high, and in 
cross-section 8 X 12 inches: what weight will it safely sus- 
tain? The height of this post, 12 x 6=: 72 inches, which is 
less than 10 x 8 (the size of the narrowest side) =: 80 inches; 
it therefore belongs to the class coming under this rule. The 
area = 8 x 12 = 96 inches ; this multiplied by 9500, the value 
of C, in the table, set opposite Georgia pine, and divided by 
6, as a factor of safety, the quotient, 152000, is the weight in 
pounds required. It will be observed that the weight would 
be the same for a Georgia-pine post of any height less than 



90 CONSTRUCTION. 

ID times 8 inches = 80 inches == 6 feet 8 inches, provided its 
breadth and thickness remain the same, 12 and 8 inches. 

(07.— Area of Post. — To find the area of the cross-sec- 
tion of a post to sustain a given weight safely, the height of 
the post being less than ten times the diameter if round, or 
ten times the least side if rectangular, the pressure coinciding 
with the axis, Ave have — 

Rule VI. — Divide the given weight in pounds by the 
value of C, in Table I. ; multiply the quotient by the factor 
of safety, and the product will be the required area in 
inches ; or — 

A = 'IJ'-. (6.) 

Example. — A weight of 40,000 pounds is to be sustained 
by a white-pine post 4 feet high : what must be its area of 
section to sustain the weight safely ? Here, 40000 divided 
by 6650, the value of C, in Table 1., set opposite white pine, 
and the quotient multiplied by 6, as a factor of safety, the pro- 
duct is 36 ; this, therefore, is the required area, and such a 
post may be 6 x 6 inches. To find the least side, so that it 
shall not be less than one tenth of the height, divide the 
height, reduced to inches, by 10, and make the least side to 
exceed this quotient. The area divided by the least side so 
determined will give the wide side. If, however, by this 
process, the first side found should prove to be the greatest, 
then the size of the post is to be found by Rule IX., X., or 
XI. 

108. — Area of Round Posts. — In case the post is to be 
round, its diameter may be found by reference to the Table 
of Circles in the Appendix, in the column of diameters, op- 
posite to the area of the post to be found in the column of 
areas, or opposite to the next nearest area. For example, 
suppose the required area, as just found by the example 
under Rule VI., is 36 : by reference to the column of areas, 
35-78 is the nearest to 36, and the diameter set opposite is 



CRUSHING STRENGTH OF SLENDER POSTS. 9 1 

6.75, which is a trifle too small. The post may therefore be, 
sa}^, 6J inches diameter. 

109. — SBender Posts. — When the height of a post is less 
than ten times its diameter, the resistance of the "post to 
crushing is approximately in proportion to its area of cross- 
section. But when the height is equal to or more than ten 
diameters, the resistance per square inch is diminished. The 
resistance diminishes as the height is increased, the diameter 
remaining the same (^Transverse Strains, Art. 643). The 
strength of a slender post, consists in a combination of the 
resistances of the material to bending and to crushing, and 
is represented in the following rule : 

110. — The Liinilt of liVeiglij. — To ascertain the weight 
that can be sustained safely by a post the height of Avhich 
IS at least ten times its least side if rectangular, or ten times 
its diameter if round, the direction of the pressure coincid- 
ing with the axis, we have — 

RiileYW, — Divide the height of the post in inches by the 
diameter, or least side, in inches ; multiply the quotient by 
itself, or take its square ; multiply the square by the value 
of e, in Table III., set opposite the kind of material of which 
the po^ is made ; to the product add the half of itself ; to 
the sum add unity (or one) ; multiply this sum by the factor 
of safety, and reserve the product for use, as below. Now 
multiply the area of cross-section of the post in inches by 
the value of C, in Table I., set opposite the material of the 
post, and divide the product by the above reserved product; 
the quotient will be the required weight in pounds ; or — 

J^-Vr^n (7-) 

Example : A Round Post. — What weight may be safely 
placed upon a post of Georgia pine 10 inches diameter and 
10 feet high, the pressure coinciding with the axis of the 
post ? The height of the post, (10 x 12 =) 120 inches, divided 
by 10, its diameter, gives a quotient of 12; this multiplied 



92 CONSTRUCTION'. 

by itself gives 144, its square; and this by -00109, the value 
of c for Georgia pine, in Table III., gives • 15696 ; to whicli 
adding its half, the sum is 0-23544; to which adding unity, 
the sum is 1-23544 ; and this multiplied by 7, as a factor of 
safety, the product is 8 -648, the reserved divisor. Now the 
area of the post is (see Table of Areas of Circles, in the Ap- 
pendix, opposite its diameter, 10)78-54; this multiplied by 
9500, the value of C for Georgia pine, in Table I., gives a 
product of 746130; which divided by 8-648, the above re- 
served divisor, gives a quotient of 86278, the required weight 
in pounds. 

AnotJicr Example : A Rectangular Post. — What weight may 
be safely placed upon a white-pine post 10 x 12 inches, and 
15 feet high, the pressure coinciding with the axis of the 
post? Proceeding according to the rule, we find the height 
of the post to be 180 inches, which divided by 10, the least 
side of the post, gives 18 ; this multiplied by itself gives 324^ 
its square ; which multiplied by -0014, the value of e for 
white pine, in Table III., gives -4536; to which adding its 
half, the sum is -6804; to which adding unity, the sum is 
I -6804 ; and this multiplied by 8, as a factor of safety, the pro- 
duct is 13-4432, the reserved divisor. Now the area of the 
post, (10 X 12 =) 120 inches, multiplied by 6650, the value of 
C for white pine, in Table I., gives a product of 798,000, and 
this divided by 13-4432, the above reserved divisor, the quo- 
tient, 59360, is the required Aveight in pounds. 

Ml. — Diameter of tlic Post: Avlien Round. — To ascertain 
the size of a round post to sustain safely a given weight, 
when the height of the post is at least ten times the diameter ; 
the direction of the pressure coinciding with the axis of the 
post; we have — 

RuleWW. — Multiply the given weight by the factor of 
safety, and divide the product by i • 5708 times the value of C 
for the material of the post, found in Table I. ; reserve the 
quotient, calling its value G. Now multiply 432 times the 
value of e for the material of the post, found in Table III., 
by the square of the height in feet, and by the above quo- 
tient G\ to the product add the square of G\ extract the 



SIZE OF POST FOR GIVEN WEIOHT. 93 

square root of the sum, and to it add the value of (j \ then 
the square root of this sum will be the required diameter; 

or — 

1.5708 6 ^ ' 



•' =\' |. 



/432 G c r' + G' ^- G 



(9.) 



Example. — What should be the diameter of a locust post 
10 feet high to sustain safely 40,000 pounds, the pressure 
coinciding with the axis? Proceeding by the rule, the given 
weight multiplied by 6, taken as a factor of safety, equals 
240000. Dividing this by 1-5708 times 11700, the value of 
6" for locust, in Table I., the quotient, 13-06, is the value of 
G, the square of which is 170-53. Now, the value of e for 
locust, in Table III., is -0015. This multiplied by 432, by 
100, the square of the height, and by the above value of G, 
gives a product of 846-2 ; w^hich added to 170-53, the above 
square of G, gives the sum of 1016-73. To 31-89, the square 
root of this, add the above value of G\ then ^-j, the square 
root of this sum, is the required diameter of the post. The 
post therefore requires to be 6-^, say 6|- inches diameter. 

112. — Side of tUe Post: ^vliesi Square.— To ascertain the 
side of a square post to sustain safely a given weight, w^hen 
the height of the post is at least ten times the side ; the pres- 
sure coinciding with the axis ; we have — 

Rule IX. — Multiply the given Aveight by the factor of 
safety, and divide the product by twice the value of C for 
the material of the post, found in Table I. ; reserve the quo- 
tient, calling its value G. Now multiply 432 times the value 
of e for the material of the post, found in Table III., by the 
square of the height in feet, and by the above quotient G\ 
to the product add the square of G ; extract the square root 
of the sum, and to it add the value of G ; then the square 
root of this sum will be the required side of the post ; or — - 

G=--^. (,o.) 



94 CONSTRUCTION. 

S =i/ . /I^gVI^TJ' + G. ^^ ^•) 



J//432 



Example. — What should be the side of a Georgia-pine 
square post 15 feet high to sustain safely 50,000 pounds, the 
pressure coinciding with the axis of the post? Proceeding 
by the rule, 50,000 pounds multiplied by 6, as a factor of 
safety, gives 300000 ; this divided by 2 x 9500 (the value of 
(7)= 19000, the quotient, 15-789, is the value of G. The 
value of € for Georgia pine is -00109; the square of the 
height is 225 ; then, 432 times -00109 by 225 and by 15-789 
(the above value of G) gives a product of 1672 - 86 ; the square 
of 6^ equals 249-31 ; this added to 1672-86 gives a sum of 
1922- 17, the square root of which is 43-843 ; which added to 
15-789, the value of G, gives 59-632, the square root of which 
is 7-722, the required side of the post. The post, therefore, 
requires to be, say, 7f inches square. 

113. — Tliickiaess of a Rectangular Post.— This may be 
definitely ascertained when the proportion which the thick- 
ness shall bear to the breadth shall have been previously 
determined. For example, when the proportion is as 6 to 8, 
then I J times 6 equals 8, and the proportion is as 1 to i^; 
again, when the proportion is as 8 to 10, then i^ times 8 
equals 10, and in this case the proportion is as i to ij. Let 
the latter figure of the ratio i to ij, i to i J, etc., be called 
11, or so that the proportion shall be as i to n, then — 

To ascertain the thickness of a post to sustain safely a 
given Aveight, when the height is at least ten times the thick- 
ness ; the action of the weight coinciding with the axis ; Vv'c 
have — 

Ride X. — Multiply the given weight by the factor of 
safety, and divide the product by twice the value of C for 
the material of the post, found in Table I., multiplied by ;/, 
as above explained ; reserve the quotient, calling it G. Now 
multiply 432 times the value of e for the material of the post, 
found in Table III., by the square of the height in feet, and 
by the above quotient G ; to the product add the square of 
G ; extract the square root of the sum, and to it add the value 



liREAUTH OF POST VOR GIVEN THICKNESS. 95 

of G\ then the square root of this sum will be the required 
thickness of the post ; or — 

0=1'^: (-) 



2 C n 



= /v 



432 G c r ^ G' -V G. (13.) 



Example. — What should be the thickness of a white-pine 
rectangular post 20 feet high to sustain safely 30,000 pounds, 
the pressure coinciding with the axis, and the proportion 
between the thickness and breadth to be as 10 to 12, or as i 
to I -2 ? Proceeding according to the rule, we have the pro- 
duct of 30000, the given weight, by 6, as a factor of safety, 
equals 180000 ; this divided by twice Cx n, or 2 x 6650 x i -2, 
(=15960) gives a quotient of 11-278, the value of G. Then, 
we have r = -0014, the square of the height equals 400; 
therefore, 432 X -0014 x 400 x 1 1 -278 = 2728-43. To this add- 
ing 127-2, the square of G^ we have 2855 63, the square 
root of which is 53-438; and this added to G gives 64.-^16, 
the square root of which is 8-045, the required thickness of 
the post. Now, since the thickness is in proportion to the 
breadth as i to 1-2, therefore 8-045 x 1-2 = 9-654, the re- 
quired width. The post, therefore, may be made 8x9! 
inches. 

114.— Breadth of a Rcctang^ular Post.— When the thick- 
ness of a post is fixed, and the breadth required ; then, to 
ascertain the breadth of a rectangular post to sustain safely 
a given weight, the direction of the pressure of which coin- 
cides with the axis of the post, we have — 

Rule XI. — Divide the height in inches by the given thick- 
ness, and multiply the quotient by itself, or take its square ; 
multiply this square by the value of ^ for the material of the 
post, found in Table 111. ; to the product add its half, and to 
the sum add unity ; multiply this sum by the given weight, 
and by the factor of safety ; divide the product b}^ the pro- 
duct of the given thickness multiplied by the value of C for 



96 CONSTRUCTION. 

the material of the post, found in Tabic I., and the quotient 
will be the required breadth ; or — 

Example. — What should be the breadth of a spruce post 
18 feet high and 6 inches thick to sustain safely 25,000 
pounds, the pressure coinciding with the axis of the post? 
According to the rule, 216 (= 12 x 18), the height in inches, 
divided by 6, the given thickness, gives a quotient of 36, the 
square of which is 1296; the value of ^' for spruce is -00098 ; 
this multiplied by 1296, the above square, equals i -27 ; which 
increased by -635, its half, amounts to 1-905 ; this increased 
by unity, the sum is 2-905 ; which multiplied by the given 
weight, and by the factor of safety, gives a product of 
435749; and this divided by 6 (the given thickness) times 7850 
(the value of (7 for spruce) = 47100, gives a quotient of 9-2516, 
the required breadth of the post. The post, therefore, re- 
quires to be 6 X 9|- inches. 

Observe that when the breadth obtained by the rule is 
less than the given thickness, the result shows that the con- 
ditions of the case arc incompatible with the rule, and that 
a new computation must be made ; taking now for the 
breadth what was before understood to be the thickness, 
and proceeding in this case, by Rule X., to find the thickness. 

115. — Re§istancc lo Ten§ion.— In Art. 95 are recorded the 
results of experiments made to test the resistance of vari- 
ous materials to tensile strain, showing in each case the ca- 
pability to such resistance per square inch of sectional area. 
The action of materials in resisting a tensile strain is quite 
simple ; their resistance is found to be directly as their sec- 
tional area. Hence — 

116. — The Umit of Wcijj^ht.— To ascertain the weight or 
pressure that may be safely applied to a beam or rod as a 
tensile strain, we have — 

Ride XII, — Multiply the area of the cross-section of the 
beam or rod in inches by the value of T, Table II.; divide 



AREA OF BEAM FOR TEx\SILE STRAIN. 97 

the product by the factor of safety, and the quotient will be 
the required weight in pounds ; or — 

W=-^^. (.5-) 



The cross-section here intended is that taken at the small- 
est part of the beam or rod. A beam, in framing, is usually 
cut with mortices ; the area will probably be smallest at the 
severest cutting ; the area used in the rule must be that of the 
uncut fibres only. 

Example. — The tie-beam of a roof-truss is of white pine, 
6 X 10 inches; the cutting for the foot of the rafter reduces 
the uncut area to 40 inches : what amount of horizontal thrust 
from the foot of the rafter will this tie-beam safely sustain ? 
Here 40 times 12000, the value of T, equals 480000; this 
divided by 6, as a factor of safety, gives 80000, the required 
weight in pounds. 

1(7. — ISectional Area. — To ascertain the sectional area of 
a beam or rod that will sustain a given weight safely, when 
applied as a tensile strain, we have — 

Ride XIII. — Multiply the given weight in pounds by the 
factor of safety ; divide the product by the value of T, Table 
II., and the quotient will be the area required in inches; 
or — 

This is the area of uncut fibres. If the piece is to be cut 
for mortices, or for any other purpose, then for this an 
adequate addition is to be made to the result found by the 
rule. 

Example. — A rafter produces a thrust horizontally of 80,000 
pounds ; the tie-beam is to be of oak : what must be the 
area of the cross-section of the tie-beam in order to sustain 
the rafter safely ? The given weight, 80000, multiplied by 
10, as a factor of safety, gives 800000; this divided by 19500, 
the value of T, Table II., the quotient, 41, is the area of uncut 
fibres. This should have usually one half of its amount 



98 CONSTRUCTION. 

added to it as an allowance for cutting; therefore, 41+21 
= 62. The tie-beam may be 6 x 10^ inches. 

Another Example. — A tie-rod of American refined 
wrought iron is required to sustain safely 36,000 pounds: 
what should be its area of cross-section ? Taking 7 as the 
factor of safety, 7 x 36000 = 252000; and this divided by 
60000, the value of T, Table II., gives a quotient of 4-2 inches, 
the required area of the rod. 

I!8. — TTeiglit of the Suspending Piece Included.— Pieces 

subjected to a tensile strain are frequently suspended verti- 
cally. In this case, at the upper end, the strain is due not 
only to the w^eight attached at the lower end, but also to the 
weight of the rod itself. Usually, in timber, this is small 
in comparison with the load, and may be neglected ; 
although in very long timbers, and where accuracy is decid- 
edly essential, as, also, when the rod is of iron, it may form 
a part of the rule. Taking the effect of the weight of the 
beam into account, the relation existing between the Aveights 
and the beam requires that the rule for the weight should 
be as follows : 

Rule XIV. — Divide the value of T for the material of the 
beam or rod, Table II., by the factor of safety ; from the 
quotient subtract 0-434 times the specific gravity of the ma- 
terial in the beam or rod multiplied by the length of the 
beam or rod in feet ; multiply the remainder by the area of 
cross-section in inches, and the product will be the required 
weight in pounds ; or — 



W: 



A \~j -0-434/^). (i;-) 



N. B. — This rule is based upon the condition that the sus- 
pending piece be not cut by mortices or in any other way. 

Example. — What weight may be safely sustained by a 
white-pine rod 4x6 inches, 40 feet long, suspended verti- 
cally? For white pine the value of T is 12000; this divid- 
ed by 8, as a factor of safety, gives 1500; from which sub^ 
tracting 0-434 times 0-458 (the specific gravity of white pine. 
Table II.) multiplied by 40, the length in feet, the remainder 



RESISTANXE TO TRANSVERSE STRAINS. 99 

is 1492-049; which multiplied by 24 (=4x6, the area of 
cross-section) equals 35,761 pounds, the required weight to be 
carried. The weight which the rule would give, neglecting 
the weight of the rod, would have been 36000; ordinarily, 
so slight a difference would be quite unimportant. 

1(9. — Area of Suspending^ Piece. — To ascertain the area 
of a suspended rod to sustain safely a given weight, when 
the weight of the suspending piece is regarded, we have — 

Rule XV. — Multiply 0-434 times the specific gravity of 
the suspending piece by the length in feet ; deduct the pro- 
duct from the quotient arising from a division of the value 
of T, Table II., by the factor of safety, and with the remain- 
der divide the given weight in pounds ; the quotient will be 
the required area in inches ; or — 

A ^ 



T , ' (i8.> 

— -0-434/^ ^ ^ 

a 

N.B. — This rule is based upon the condition that the rod 
be not injured in anywise by cutting. 

Example. — What should be the area of a bar of English 
cast iron 20 feet long to sustain safely, suspended from its 
lower end, a weight of 5000 pounds ? Taking the factor of 
safety at 7-0, and the specific gravity also at 7, and the 
value of T, Table II., at 17000, we have the product of 
0-434 X 7.0 X 20 = 60-76; then 17000 divided by 7 gives 
a quotient of 2428-57; from which deducting the above 
60-76, there remains 2367-81 ; dividing 5000, the given 
weight, by this remainder, we have the quotient, 2 • 1 1, which 
is the required area in inches. 

RESISTANCE TO TRANSVERSE STRAINS. 

120. — Transverse Strains: Rupture. — A load placed 
upon a beam, laid horizontally or inclined, will bend it, and, 
if the weight be proportionally large, will break it. The 
power in the material that resists this bending or breaking 
is termed the resistance to cross-strains, or transverse strains. 



lOO CONSTRUCTION. 

While in posts or struts the material is compressed or short- 
ened, and in ties and suspending pieces it is extended or 
lengthened, in beams subjected to cross-strains the material 
is both compressed and extended. (See Art. 91.) When the 
beam is bent the fibres on the concave side are compressed, 
while those on the convex side are extended. The line 
where these two portions of the beam meet — that is, the 
portion compressed and the portion extended — the hori- 
zontal line of juncture, is termed the neutral line or plane. 
It is so called because at this line the fibres are neither com- 
pressed nor extended, and hence are under no strain what- 
ever. The location of this line or plane is not far from the 
middle of the depth of the beam, when the strain is not suf- 
ficient to injure the elasticity of the material ; but it re- 
moves towards the concave or convex side of the beam as 
the strain is increased, until, at the period of rupture, its 
distance from the top of the beam is in proportion to its dis- 
tance from the bottom of the beam as the tensile strength of 
the material is to its compressive strength. 

(21. — Liocatioii of Mortices.— In order that the diminution 
of the strength of a beam by framing be as small as possible, 
all mortices should be located at or near the middle of the 
depth. There is a prevalent idea with some, who are aware 
that the upper fibres of a beam are compressed Avhen sub- 
ject to cross-strains, that it is not injurious to cut these top 
fibres, provided that the cutting be for the insertion of an- 
other piece of timber — as in the case of gaining the ends of 
beams into the side of a girder. They suppose that the piece 
filled in will as effectually resist the compression as the part 
removed would have done, had it not been taken out. Now, 
besides the effect of shrinkage, which of itself is quite suf- 
ficient to prevent the proper resistance to the strain, there 
is the mechanical difficulty of fitting the joints perfectly 
throughout ; and, also, a great loss in the power of resist- 
ance, as the material is so much less capable of resistance 
when pressed at right angles to the direction of the fibres 
than when directly with them, as the results of the experi- 
ments in the tables show. 



STRENGTH OF BEAMS FOR CROSS-STRAINS. lOI 

122- — Transverse Strains : Relation of Weight lo I>i- 
mcnsions. — The strength of various materials, in their re- 
sistance to cross-strains, is given in Table III., Art. 96. The 
second column of the table contains the results of experi- 
ments made to test their resistance to rupture. In the case 
of each material, the figures given and represented by B 
indicate the pounds at the middle required to break a tc7tit 
of the material, or a piece i inch square and i foot long 
between the bearings upon which the piece rests. To be 
able to use these indices of strength, in the computation of 
the strength of large beams, it is requisite, first, to establish 
the relation between the unit of material and the larger 
beam. Now, it may be easily comprehended that the strength 
of beams will be in proportion to their breadth ; that is, 
Avhen the length and depth remain the same, the strength 
will be directly as the breadth ; for it is evident that a beam 
2 inches broad will bear twice as much as one which is only 
I inch broad, or that one which is 6 inches broad will bear 
three times as much as one which is 2 inches broad. Thij; 
establishes the relation of the weight to the breadth. With 
the depth, however, the relation is different ; the strength is 
greater than simply in proportion to the depth. If the 
boards cut from a squared piece of timber be piled up in 
the order in which they came from the timber, and be loaded 
with a heavy weight at the middle, the boards will deflect 
or sag much more than they would have done in the timber 
before sawing. The greater strength of the material when 
in a solid piece of timber is due to the cohesion of the fibres 
at the line of separation, by which the several boards, before 
sawing, are prevented from sliding upon each other, and 
thus the resistance to compression and tension is made to 
contribute to the strength. This resistance is found to be 
in proportion to the depth. Thus the strength due to the 
depth is, first, that which arises from the quantity of the 
material (the greater the depth, the more the material), 
which is in proportion to the depth ; then, that which en- 
sues from the cohesion of the fibres in such a manner as to 
prevent sliding ; this is also as the depth. Combining the 
two, we have, as the total result, the resistance in proportion 



I02 CONSTRUCTION. 

to the square of the depth. The relation between the 
weight and the length is such that the longer the beam is, 
the less it will resist ; a beam which is 20 feet long will sus- 
tain only half as much as one which is 10 feet long ; the 
breadth and depth each being the same in the two beams. 
From this it results that the resistance is inversely in pro- 
portion to the length. To obtain, therefore, the relation 
between the strength of the unit of material and that of a 
larger beam, we have these facts, namely : the strength of 
the unit is the value of j5, as recorded in Table III. ; and 
the strength of the larger beam, represented by W, the 
Aveight required to break it, is the product of the breadth 
into the square of the depth, divided by the length ; or, 
while for the unit we have the ratio — 

B\ I, 

we have for the larger beam the ratio — 

Therefore, putting these ratios in an expressed proportion, 
we have — 

^: I :: W\^-^l 

From which (the product of the means equalling the pro- 
duct of the extremes ; see Art, 373) we have — 

lV=-j-. (19.) 

In which JV represents the pounds required to break a 
beam, when acting at the middle between the two supports 
upon which the beam is laid ; of which beam ^ represents 
the breadth and <^ the depth, both in inches, and / the length 
in feet between the supports ; and B is from Table III., and 
represents the pounds required to break a unit of material 
like that contained in the larger beam. 



IJMIT OF WEIGHT AT MIDDLE. IO3 

(23- — Safe IVeight : Load at Hiddle. — The relation 
established, in the last article, between the weight and the 
dimensions is that which exists at the moment of rupture. 
The rule (19.) derived therefrom is not, therefore, directly 
practicable for computing the dimensions of beams for 
buildings. From it, however, one may readily be deduced 
which shall be practicable. In the fifth column of Table III. 
are given the least values of ^, the factor of safety, explained 
in Art. 96. Now, if in place of By the symbol for the break- 
ing weight, the quotient of B divided by a be substituted, 
then the rule at once becomes practicable ; the results now 
being in consonance with the requirements for materials 
used in buildings. Thus, with this modification, we have — 

W — . — -— • (20.) 

a I 

Therefore, to ascertain the weight which a beam may be 
safely loaded with at the centre, we have — 

Rule XVI.— Multiply the value of B, Table III., for the 
kind of material in the beam by the breadth and by the 
square of the depth of the beam in inches ; divide the pro- 
duct by the product of the factor of safety into the length 
of the beam between bearings in feet, and the quotient will 
be the weight in pounds that the beam will safely sustain 
at the middle of its length. 

Example. — What weight in pounds can be suspended 
safely from the middle of a Georgia-pine beam 4x lo inches, 
and 20 feet long between the bearings ? For Georgia pine the 
value of B, in Table III., is 850, and the least value of a is 
1-84. For reasons given in Art. 96, let a be taken as high 
as 4; then, in this case, the value of b is 4, and that of d is 
10, while that of/ is 20. Therefore, proceeding by -the rule, 
850 X 4 X 10' = 340000 ; this divided by 4 x 20 (= 80) gives 
a quotient of 4250 pounds, the required weight. 

Observe that, had the value of a been taken at 3, instead 
of 4, the result by the rule would have been a load of 5667 
pounds, instead of 4250, and the larger amount w^ould be 
none too much for a safe load upon such a beam ; although, 



I04 CONSTRUCTION. 

with it, the deflection would be one third greater than with 
the lesser load. The value of a should always be assigned 
higher than the figures of the table, which show it at its 
least value ; but just how much higher must depend upon 
the firmness required and the conditions of each particular 
case. 

(24-. — Breadth of Beam witli Safe Load. — By a simple 
transposition of the factors in equation (20.\ we obtain — 

a rule for the breadth of the beam. 

Therefore, to ascertain what should be the breadth of a 
beam of given depth and length to safely sustain at the 
middle a given weight, we have — ■ 

RuL' XVII. — Multiply the given Aveight in pounds by 
the factor of safety, and by the length in feet, and divide the 
product by the square of the depth multiplied by the value 
of B for the material in the beam, in Table III. ; the quotient 
will be the required breadth. 

Example. — What should be the breadth of a white-pine 
beam 8 inches deep and lo feet long between bearings to 
sustain safely 2400 pounds at the middle? For white pine 
the value of B, in Table III., is 500. Taking the value of a 
at 4, and proceeding by the rule, we have 2400 x4x 10 = 
96000 ; this divided by i^^ x 500 =) 32000 gives a quotient 
of 3, the required breadth of the beam. 

826. — Beptli of Beam Avith §afe Load. — A transposition 
of the factors in equation (21.), and marking it for extraction 
of the square root, gives — 



B 

2i rule for the depth of a beam. Therefore, to ascertain what 
should be the depth of a beam of given breadth and length 
to safely sustain a given weight at the middle, we have — 



WEKillT AT ANY POINT. I05 

Ride XVIII. — Multiply the given weight by the factor of 
safety, and by the length in feet; divide the product by the 
product of the breadth into the value of B for the kind of 
wood, Table III. ; then the square root of the quotient will 
be the required depth. 

Example. — What should be the depth of a spruce beam 
5 inches broad and 10 feet long between bearings to sustain 
safely, at middle, 4500 pounds ? The value of B from the table 
is 550; taking a at 4, and proceeding by the rule, we have 
4500 X 4x15 = 2700CO ; this divided by (550 x 5 ==) 2750 
gives a quotient of 98-18, the square root of which is 9-909, 
the required depth of the beam. The beam should be 5 x 10 
inches. 

126- — Safe Load at any Point. — When the load is at the 
middle of a beam it exerts the greatest possible strain ; at 
any other point the strain would be less. The strain de- 
creases gradually as it approaches one of the bearings, and 
when arrived at the bearing its effect upon the beam as a 
cross-strain is zero. The effect of a weight upon a beam is 
in proportion to its distance from one of the bearings, mul- 
tiplied by the portion of the load borne by that bearing. 

The load upon a beam is divided upon the two bearings, 
a3 shown at Art. 88. The weight which is required to rup- 
ture a beam is in proportion to the breadth and square of 
the depth, b d', as before shown, and also in proportion to 
the length divided by 4 times the rectangle of the two parts 

into which the load divides the lenorth, or- ^ (see Fi^^. 35). 

This, when the load is at the middle, may be put as 

= ---, a result coincidinjr with the relation before 



4xi/x-^/ / 

given in Art. 122, viz. : "The resistance is inversely in pro- 
portion to the length." The total resistance, therefore, put- 
ting the two statements together, is in proportion to . 

There are, therefore, the^e two ratios, viz., JV: — - — - and 

4 7/1 n 

B : \^ from which vjc have the proportion — 



I06 CONSTRUCTION. 

/.'::::/F:^^, 
4 m 11 

from which we have — 



1V= . (23.) 

4 ;;/ 71 

r> 

This is the relation at the point of rupture, and when — is 

^ ^ a 

used instead of B, the expression gives the safe weight. 

Therefore — 

7F=_^Mlf (24.) 

4 a 7/1 71 

is an expression for the safe Aveight. Now, to ascertain the 
weight which may be safely borne by a beam at any point 
in its length, we have — 

Rule XIX. — Multiply the breadth by the square of the 
depth, by the length in feet, and by the value of B for the 
material of the beam, in Table III. ; divide the product by 
the product of four times the factor of safety into the rec- 
tangle of the two parts into which the centre of gravity of 
the weight divides the beam, and the quotient will be the 
required weight in pounds. 

Exai7tph\ — What weight may be safely sustained at 3 feet 
from one end of a Georgia-pine beam which is 4 x 10 inches, 
and 20 feet long? The value of B for Georgia pine, in 
Table III., is 850 ; therefore, by the rule, 4 x lo"^ x 20 x 850 = 
6800000. Taking the factor of safety at 4, we have 
4x4x3x17=816. Using this as a divisor with which to 
divide the former product, we have as a quotient 8333 
pounds, the required weight. 

127. — Breadtli or Depth: Load at any Point. — By a 

proper transposition of the factors of (24.) we obtain — 

, .2 4 W a 171 71 . . 

hd =^y-, 125.) 

an expression showing the product of the breadth into the 
square of the depth ; hence, to ascertain the breadth or 



DISTRIBUTED WEIGHT. 10/ 

depth of a beam to sustain safely a given weight located at 
any point on the beam, wc have — 

Rule XX. — Multiply four times the given weight by the 
factor of safety, and by the rectangle of the two parts into 
which the load divides the length ; divide the product by 
the product of the length into the value of B for the mate- 
rial of the beam, found in Table III., and the quotient will be 
equal to the product of the breadth into the square of the 
depth. Now, to obtain the breadth, divide this product by 
the square of the depth, and the quotient will be the required 
breadth. But if, instead of the breadth, the depth be de- 
sired, divide the said product by the breadth ; then the 
square root of the quotient Avill be the required depth. 

Example. — What should be the breadth (the depth being 
8) of a white-pine beam 12 feet long to safely custain 3500 
pounds at 3 feet from one end ? Also, what should be its 
depth when the breadth is 3 inches? By the rule, taking 
the factor of safety at 4, 4 x 3500 x 4 x 3 x 9 = 15 12000. 
The value of B for white pine, in Table III., is 500 ; there- 
fore, 500 X 12 = 6000; with this as divisor, dividing 15 12000, 
the quotient is 252. Now, to obtain the breadth when the 
depth is 8, 252 divided by (8x8=) 64 gives a quotient of 
3.9375, the required breadth; or the beam may be, say, 4x8. 
Again, when the breadth is 3 inches, we have for the quotient 
of 252 divided by 3 = 84, and the square root of 84 is 9- 165, 
or 9jr inches. For this case, therefore, the beam should be, 
say, 3 X 9J inches. 

120. — ^Veiglit UnBformly Di§tribme4l. — When the load is 
spread out uniformly over the length of a beam, the beam 
will require just twice the weight to break it that would be 
required if the weight were concentrated at the centre. 

Therefore, we have JF= — , where U represents the dis- 
tributed load. Substituting this value of W in equation 
(20.), we have — 

U_BbjP 

2 a I ' 

or — 

£/=---_-. (36.) 



I08 CONSTRUCTION. 

Therefore, to ascertain the weight which may be safely sus- 
tained, when uniformly distributed over the length of a 
beam, wc have — 

Rule XXL — Multiply twice the breadth by the square of 
the depth, and by the value of B for the material of the 
beam, in Table III., and divide the product by, the product 
of the length in feet by the factor of safety, and the quotient 
will be the required weight in pounds. 

Example. — What weight uniformly distributed may be 
safely sustained upon a hemlock beam 4x9 inches, and 20 
feet long? The value of B for hemlock, in Table III., is 
450 ; therefore, by the rule, 2 x 4 x 9' x 450 = 291 600. Tak- 
ing the factor of safety at 4, we have 4 x 20 = 80, the pro- 
duct by which the former product is to be divided. This 
division produces a quotient of 3645, the required weight. 

129.— Breadth or Oeptli : Load Uniformly Distributed.— 

By a proper transposition of factors in (26.), we obtain — 

, ,2 Ual , . 

bd^ = ~^^, (27.) 

an expression giving the value of the breadth into the square 
of the depth. From this, therefore, to ascertain the breadth 
or the depth of a beam to sustain safely a given weight uni- 
formly distributed over the length of a beam, we have — 

Rule XXII. — Multiply the given w- eight by the factor of 
safety, and by the length ; divide the product by the pro- 
duct of twice the value of B for the material of the beam, 
in Table III., and the quotient will be equal to the breadth 
into the square of the depth. Now, to find the breadth, 
divide the said quotient by the square of the depth ; but if, 
instead of the breadth, the depth be required, then divide 
said quotient by the breadth, and the square root of this 
quotient will be the required depth. 

Example. — What should be the size of a white-pine beam 20 
feet long to sustain safely 10,000 pounds uniformly distributed 
over its length? The value of ^ for white pine, in Table III., 
is 500. Let the factor of safety be taken at 4. Then, by the 
rule, loooo x 4 x 20 = 800000; this divided by (2 x 500 =) 



WEIGHT PER BEAM IN FLOORS. 1 09 

a quotient of 800. Now, if the depth be fixed at 
12, then the said quotient, 8oo, divided by (12 x 12=) 1.44 
gives 5-y, the required breadth of beam ; and the beam may 
be, say, 5} x 12. Again, if the breadth is fixed, say, at 6, and 
the depth is required, then the said quotient, See, divided by 
6 gives 1 33 J, the square root of which, 11 • 55, is the required 
depth. The beam in this case should therefore be, say, 
6 X lif inches. 

130. — ILoad per Foot Superficial. — When several beams 
are laid in a tier, placed at equal distances apart, as in a tier 
of floor-beams, it is desirable to know what should be their 
size in order to sustain a load eqiially distributed over the 
floor. ,^.--^ 

If the distance apart at which they are placed, measured 
from the centres of the beams, be multiplied by the length 
of the beams between bearings, the product will equal the 
area of the floor sustained by one beam ; and if this area be 
multiplied by the weight upon a superficial foot of the floor, 
the product will equal the total load uniformly distributed 
over the length of the beam ; or, if c be put to represent the 
distance apart between the centres of the beams in feet, and 
/represent the length in feet of the beam between bearings, 
and/ equal the pounds per superficial foot on the floor, 
then the product of these, or <://, will represent the uni- 
formly distributed load on a beam ; but this load was before 
represented by U {Art. 128); therefore, we have cfl— U^ 
and they may be substituted for it in (26.) and (27.). Thus 
we have — 



2 B 



or- 



bd-"^ -'^~-. (28.) 

2B 

Therefore, to ascertain the size of floor-beams to sustain 
safely a given load per superficial foot, we have — 

7?///^ XXIII. — Multiply the given weight per superficial 
foot by the factor of safety, by the distance between the 



no CONSTRUCTION. 

centres of the beams in feet, and by the square of the length 
in feet; divide the product by twice the value of B for the 
material of the beams, in Table III., and the quotient will be 
equal to the breadth into the square of the depth. Now, to 
obtain the breadth, divide said quotient by the square of the 
depth, and this quotient will be the required breadth. But 
if, instead of the breadth, the depth be required, divide the 
aforesaid quotient by the breadth ; then the square root of 
this quotient will be the required depth. 

Example. — What should be the size of white-pine floor- 
beams 20 feet long, placed 16 inches from centres, to sustain 
safely 90 pounds per superficial foot, including the weight 
of the materials of construction — the beams, flooring, plas- 
tering, etc.? The value of B for white pine is 500; the 
factor of safety may be put at 5. Then, by the rule, we 
have 90 X 5 X 11- X 20^ = 240000. This divided by (2 x 500 
= ) 1000 gives 240. Now, for the breadth, if the depth be 
fixed at 9 inches, then 240 divided by (9^ = ) 81 gives a 
quotient of 2-963. The beams therefore should be, say, 
3x9. But if the breadth be hxed, say, at 2-5 inches, then 
240 divided by 2-5 gives a quotient of 96, the square root of 
which is 9-8 nearly. The beams in this case would require 
therefore to be, say, 2|- x 10 inches. 

N. B. — It is well to observe that the question decided 
by Rule XXII. is simply that of strength only. Floor-beams 
computed by it will be quite safe against rupture, but they 
will in most cases deflect much more than Avould be consist- 
ent with their good appearance. Floor-beams should be 
computed by the rules which include the effect of deflection. 
(See Art, 152.) 

131. — L.evcrs : I^oad at One End. — The beams so far con- 
sidered as being exposed to transverse strains have been 
supposed to be supported at each end. When a piece is 
held firmly at one end only, and loaded at the other, it is 
termed a lever ; and the load which a piece so held and 
loaded will sustain is equal to one fourth that which the 
same piece would sustain if it were supported at each end 
and loaded at the middle. Or, the strain in a beam sup- 



LEVERS TO SUSTAIN GIVEN WEIGHTS. I I I 

ported at each end caused by a given weight located at the 
middle is equal to that in a lever of the same breadth and 
depth, when the length of the latter is equal to one half that 
of the beam, and the load at its end is equal to one half of 
that at the middle of the beam. Or, when P represents the 
load at the end of the lever, and n its length, then W =^ 2 P, 
and l—2n. Substituting these values of [Fand / in equa- 
tion (20.), we have — 



.P^ 


Bbd' 
2 an 


p^ 


Bbd' 



from which- 

/?/, //'^ 

(29.) 
^an 

Hence, to ascertain the weight which may be safely sus- 
tained at the end of a lever, we have — 

Rule XXIV.— Multiply the breadth of the lever by the 
square of its depth, and by the value of B for the material 
of the lever, in Table III. ; divide the product by the pro- 
duct of four times the length in feet into the factor of safety, 
and the quotient will be the required weight in pounds. 

Example. — What weight can be safely sustained at the 
end of a maple lever of which the breadth is 2 inches, the 
depth is 4 inches, and the length is 6 feet ? The value of B 
for maple, in Table III., is iioo; therefore, by the rule, 
2 X 4^" X 1 100 = 35200. And, taking the factor of safety at 5, 
4x5x6= 120, and 35200 divided by 120 gives a quotient of 
293 •33» or 293^- pounds. 

N. B. — When a lever is loaded with a weight uniformly 
distributed over its length, it will sustain just twice the load 
which can be sustained at the end. 

132. — L<ever§ : Breadth or Depth. — By a proper trans- 
position of the factors in (29.), we obtain — 

b d' = tL_ — (30.) 

Hence, to ascertain the breadth or depth of a lever to sus- 
tain safely a given weight, we have — 



112 CONSTRUCTION. 

Rule XXV. — Multiply four times the given weight by 
the length of the lever, and by the factor of safety ; divide 
the product by the value of B for the material of the lever, 
in Table III., and the quotient will be equal to the breadth 
multiplied by the square of the depth. Now, if the breadth 
be required, divide said quotient by the square of the depth, 
and this quotient will be the required breadth ; but if, 
instead of the breadth, the depth be required, divide the 
said quotient by the breadth ; then the square root of this 
quotient will be the required depth. 

Example. — What should be the size of a cherr}^ lever 5 
feet long to sustain safely 250 pounds at its end? Proceed- 
ing by the rule, taking the factor of safety at 5, we have 
4x250x5x5 = 25000. The value of B for cherry, in Table 
III., is 650; and 25000 divided by 650 gives a quotient of 
38-46. Now, if the depth be fixed at 4, then 38-46 divided 
by (4x4=) 16 gives a quotient of 2-4, the required breadth. 
But if the breadth be fixed at 2, then 38-46 divided by 2 
gives a quotient of 19-23, the square root of which is 4-38, 
the required depth. Therefore, the lever maybe 2-4x4, 
or 2x4! inches. 

J33. — Dcflec!tioii : Relation to ^VclglU. — When a load is 
placed upon a beam supported at each end, the beam bends 
more or less ; the distance that the beam descends under 
the operation of the load, measured at the middle of its 
length, is termed its deflection. In an investigation of the 
laws of deflection it has been demonstrated, and experiments 
have confirmed it, that while the elasticity of the material 
remains uninjured by the pressure, or is injured in but a 
small degree, the amount of deflection is directly in propor- 
tion to the weight producing it ; for example, if 1000 pounds 
laid upon a beam is found to cause it to deflect or descend at 
the middle a quarter of an inch, then 2000 pounds will cause 
it to deflect half an inch, 3000 pounds will deflect it three 
fourths of an inch, and so on. 

134-. — Deflection : Relation to Dimensions. — In Table 
III. are recorded the results of experiments made to test the 



THE LAW OF DEFLECTION. II3 

resistance of the materials named to deflection. The fig- 
ures in the third column designated by the letter F (for flex- 
ure) show the number of pounds required to deflect a unit 
of material one inch. This is an extreme state of the case, 
for in most kinds of material this amount of depression 
would exceed the limits of elasticity ; and hence the rule 
would here fail to give the correct relation as between the 
dimensions and pressure. For the law of deflection as above 
stated (the deflections being in proportion to the w^eights) 
is true only while the depressions are small in comparison 
with the length. Nothing useful is, therefore, derived from 
this position of the question, except to give an idea of the 
nature of the quantity represented by the constant F\ it 
being in reality an index of the stiffness of the kind of mate- 
rial used in comparing one material with another. Whatever 
be the dimensions of the beam, F will always be the same 
quantity for the same material ; but among various materials 
F will vary according to the flexibility or stiffness of each 
particular material. For example, F will be much greater 
for iron than for wood ; and again, among the various kinds 
of wood, it will be larger for the stiff woods than for those 
that are flexible. The value of F, therefore, is the weight 
which would deflect the unit of material one inch, upon the 
supposition that the deflections, from zero to the depth of 
one inch, continue regularly in proportion to the increments 
of weight producing the deflections, or, for each deflection — 

F '. W'.W \ 6^ 

from which we have — 

W 
W=Fd; or,F=!^, 
o 

in which 6 represents the deflection in inches corresponding 
to W, the weight producing it. This is for the unit of ma- 
terial. For beams of larger dimensions, investigations have 
shown {Transverse Strains, Chapters XIII. and XIV.) that 
the power of a beam to resist deflection by a weight at mid- 
dle is in proportion to its breadth and the cube of its depth, 
and it is inversely in proportion to the cube of the length ; 



114 CONSTRUCTION. 



or, when the resistance of the unit of material is measured, 
as above, by — 
larger beam of- 



W 
as above, by — -, we have the relation between it and a 



Putting this ratio in a proportion with that of the unit of 
material, we have — 

W bd' 



F ', \ '.'. -J- ' -j,- 



which gives — 



W Fbd' 



from which we have- 



6 - r ' 

^==-77— (31.) 

135. — Deflection : Weight when at Middle. — In equation 
(31.) we have a rule by which to ascertain what weight is 
required to deflect a given beam to a given depth of deflec- 
tion ; this, in words at length, is — 

Rule XXVI.— Multiply the breadth of the beam by the 
cube of its depth, and by the given deflection, all in inches, 
and by the value of F iox the material of the beam, in Table 
III.; divide the product by the cube of the length in feet, 
and the quotient will be the required weight in pounds. 

Example. — What weight is required at the middle of a 
4x 12 inch Georgia-pine beam 20 feet long to deflect it 
three quarters of an inch? The value of i^ for Georgia 
Ipine, in Table III., is 5900; therefore, by the rule, we have 
4 X 12' X 0-75 X 5900 = 30585600, which divided by (20x20 
X 20 =) 8000 gives a quotient of 3823-2, the required weight 
in pounds. 

136.— Deflection: Breadth or Depth, ^¥eig[ht at Middle. 

— By a transposition of equation (31.), we obtain — 

^"^ =-ft^ . (32.) 



SIZE FOR A GIVEN DEFLECTION. 



115 



a rule by which may be found the breadth or depth of a 
beam, with a given load at middle and with a given deflec- 
tion ; this, in words at length, is — 

Rule XXVI I. — Multiply the given load by the cube of 
the length in feet, and divide the product by the product of 
the deflection into the value of F for the material of the 
beam, in Table III. ; then the quotient will be equal to the 
breadth of the beam multiplied by the cube of its depth, 
both in inches. 

Now, to obtain the breadth, divide the said quotient by 
the cube of the depth, and this quotient will be the required 
breadth. But if, instead of the breadth, the depth be re- 
quired, then divide the said quotient by the breadth, and 
the cube root of this quotient will be the required depth. 
But if neither breadth nor depth be previously fixed, but it 
be required that they bear a certain proportion to each 
other ; such that d \ b w i : r, r being a decimal, then b ^= r d, 
and b d"" — r<^* ; then, to find the depth, divide the aforesaid 
quotient by the decimal r, and the fourth root (or the square 
root of the square root) will be the required depth, and this 
multiplied by the decimal r will give the breadth. 

Example. — What should be the size of a spruce beam 20 
feet long between bearings, sustaining 2000 pounds at the 
middle, with a deflection of one inch ? By the rule, the 
weight into the cube of the length is 2000 x 8000 = 16000000. 
The value of Fior spruce, in Table III., is 3500; this by the 
deflection = i gives 3500, which used as a divisor in divid- 
ing the above 16000000 gives a quotient of 4571 -43. Now, 
if the breadth be required, the depth being fixed, say, at 10, 
then 4571 .43 divided by (10 x 10 x 10 =) 1000 gives 4- 57, the 
required breadth. The beam should be, say, 4f by 10 inches. 
But if the depth be required, the breadth being fixed, say, at 
4, then 4571.43 divided by 4 gives 1142-86, the cube root 
of which is 10-46; so in this case, therefore, the beam is 
required to be 4xioj inches. Again, if the breadth is to 
bear a certain proportion to the depth, or that the ratio be- 
tween them is to be, say, o- 6 to i, then let r = o-6, and then 
4571-43 = o-6<^*, and dividing by 0-6, we have 7619-05 
= d\ This equals d"" xd"^ \ therefore the square root of 7619 



Il6 CONSTRUCTION. 

is 87-29, and the square root of this is 9-343, the required 
depth in inches. Now 9-343x0-6 equals the breadth, or 
9-343x0-6=5-6; therefore the beam is required to be 
5 -6 X 9- 34 inches, or, say, sf x g^ inches. 

137.— Deflection : when "Weight i§ at middle. — By a trans- 
position of the factors in (32.), we obtain — 

'=-FJ^^ (33.) 

a rule by which the deflection of any given beam may be as- 
certained, and which, in words at length, is — 

Rule XXVIII. — Multiply the given weight by the cube 
of the length in feet ; divide the product by the product of 
the breadth into the cube of the depth in inches, multiplied 
by the value of Fior the material of the beam, in Table III., 
and the quotient will be the required deflection in inches. 

Example. — To what depth will 1000 pounds deflect a 
3 X 10 inch white-pine beam 20 feet long, the weight being 
at the middle of the beam? By the rule, we have looox 20' 
= 8000000 ; then, since the value of F for white pine, in 
Table III., is 2900, we have 3 x 10' x 2900 = 8700000 ; using 
this product as a divisor and by it dividing the former pro- 
duct, we obtain a quotient of 0.9195, the required deflection 
in inches. 

138. — Deflection: Load Umformly Distributed. — In two 

beams of equal capacity, suppose the one loaded at the 
middle, and the other with its load uniformly distributed 
over its length, and so loaded that the deflection in one beam 
shall equal that in the other ; then the weight at the middle 
of the former beam will be equal to five eighths of that on 
the latter. This proportion between the two has been de- 
monstrated by writers on the strength of materials. (See p. 
484, Mechanics of Eng. and Arch., by Prof. Mosely, Am. ed. by 
Prof. Mahan, 1856.) Hence, when £/is put to represent the 
uniformly distributed load, we have — 



DEFLECTION FOR LOAD EQUALLY DISTRIBUTED. 1 17 

or, when an equally distributed load deflects a beam to a 
certain depth, five eighths of that load, if concentrated at 
the middle, would cause an equal deflection. This value of 
IV may therefore be substituted for it in equation (31.), and 
give— 



from which we obtain — 



/3 



U^ j7 » (34-) 

a rule for a uniformly distributed load. 

139. — Deflection: Weight when Uniformly Distributed. 

— In equation (34.) we have a rule by which we may ascertain 
what weight is required to deflect to a given depth any 
given beam. This, in words at length, is — 

Rule XXIX.— Multiply 1-6 times the deflection by the 
breadth of the beam, and by the cube of its depth, all in 
inches, and by the value of F ior the material of the beam, 
in Table III. ; divide the product by the cube of the length 
in feet, and the quotient will be the required weight in 
pounds. 

Example. — What weight, uniformly distributed over the 
length of a spruce beam, will be required to deflect it to the 
depth ot 0-5 ot an mch, the beam being 3 x 10 inches and 10 
feet long? The value of i^for spruce, in Table III., is 3500. 
Therefore, by the rule, we have i-6xo-5 x 3 x 10^ x 3500 = 
8400000, and this divided by (10x10x10=) 1000 gives 
8400, the required weight in pounds. 

140. — Deflection: Breadth or Depth, Load Uniformly 
Distributed. — By transposition of the factors in equation 
(34.), we obtain — 

bd^ =-^. (35.) 

a rule for the dimensions, which, in words at length, is — 



Il8 CONSTRUCTION. 

Rule XXX.— Multiply the given weight by the cube of 
the length of the beam ; divide the product by i -6 times the 
given deflection in inches, multiplied by the value of F for 
the material of the beam, in Table III., and the quotient will 
equal the breadth into the cube of the depth. Now, to ob- 
tain the breadth, divide this quotient by the cube of the depth, 
and the resulting quotient will be the required breadth in 
inches. But if, instead of the breadth, the depth be required, 
then divide the aforesaid quotient by the breadth, and the 
cube root of the resulting quotient will be the required depth 
in inches. Again, if neither breadth nor depth be previously 
determined, but to be in proportion to each other at a given 
ratio, as r to i, r being a decimal fixed at pleasure, then di- 
vide the aforesaid quotient by the value of r, and take the 
square root of the quotient; then the square root of this 
square root will be the required depth in inches. The breadth 
will equal the depth multiplied by the value of the deci- 
mal r. 

Example. — What should be the size of a locust beam lo 
feet long which is to be loaded with 6000 pounds equally 
distributed over the length, and with which the beam is to 
be deflected f of an inch ? The value of F for locust, in 
Table III., is 5050. By therule, we have6ooox(io x 10 x 10=) 
1000 = 6000000, which is to be divided by(i •6xo-75x 5050 —) 
6060, giving a quotient of 990- 1. Now, if the depth be, say, 
6 inches, then 990- 1 divided by (6 x 6 x 6 =) 216 gives a quo- 
tient of 4-584, the required breadth in inches, say 4f. But 
if the breadth be assumed at 4 inches, then 990- 1 divided by 
4 gives a quotient of 247- 525, the cube root of which is 6-279, 
the required depth in inches, or, say, 6J. And, again, if the 
ratio between the breadth and depth be as o- 7 to i, then 990- 1 
divided by 0-7 gives a quotient of 1414-43, the square root 
of which is 37-609, of which the square root is 6-1326, the 
required depth in inches, or, say, 6|- ; and then 6-1326x0-7 = 
4-293, the required breadth in inches ; or, the beam should 
be 4y\ X 6\ inches. 

14-1. — Deflection : when IVeight is Uniformly Distributed. 

— By a transposition of the factors of equation (35.), we ob- 
tain — 



DEFLECTION OF LEVERS AND BEAMS. II9 

a result nearly the same as that in equation (33.), which is a 
rule for deflection by a weight at middle, and which by 
slight modifications may be used for deflection by an equally 
distributed load. Thus by — 

Rule XXXI.— Proceed as directed in Rule XXVlU.{ArL 
137), using the equally distributed weight instead of a con- 
centrated weight, and then divide the result there obtained 
for deflection by i -6 ; then the quotient Avill be the required 
deflection in inches. 

Example. — Taking the example given under Rule 
XXVIII. , in Arl. 137, and assuming that the 1000 pounds load 
with which the beam is loaded be equally distributed, then 
0-9195, the result for deflection as there found, divided by i -6, 
as by the above rule, gives 0-5747, the required deflection. 
This result is just five eighths of 0-9195, the deflection by the 
load at middle. 

N.B. — The deflection by a uniformly distributed load is 
just five eighths of that produced by the same load when 
concentrated at the middle of the beam ; therefore, five 
eighths of the deflection obtained by Rule XXVIII. will be 
the deflection of the same beam when the same weisfht is 
uniformly distributed. 



'fe- 



14-2. — Deflection of Licvers. — The deflection of a lever is 
the same as that of a beam of the same breadth and depth, 
but of twice the length, and loaded at the middle with a load 
equal to twice that which is at the end of the lever. There- 
fore, if P represents the weight at the end of a lever, and n 
the length of the lever in feet, then 2 P= JV SLXid 2 11 — /, and 
if these values of I^Fand /be substituted for those in equa- 
tion (33.), we obtain — 

2 P X 2 ;/ ^ 



which reduces to — 



Fbd' 






I20 CONSTRUCTION. 

a result i6 times that in equation (33.), which is the deflection 
in a beam. Therefore, when a beam and a lever equal in 
sectional area and in length be loaded by equal weights, the 
one at the middle, the other at one end, the deflection of the 
lever will be 16 times that of the beam. This proportion is 
based upon the condition that neither the beam nor the lever 
shall be deflected beyond the limits of elasticity. 

143. — Deflection of a Liever: L<oad at End. — Equation 
(37.), in words at length, is — 

Rule XXXII. — Multiply 16 times the given weight by 
the cube of the length in feet ; divide the product by the 
product of the breadth into the cube of the depth multiplied 
by the value of i^for the material of the lever, in Table III., 
and the quotient will be the required deflection. 

Example. — What would be the deflection of a bar of 
American wrought iron one inch broad, two inches deep, 
loaded with 1 50 pounds at a point 5 feet distant from the 
wall in which the bar is imbedded ? The value of F for 
American wrought iron, in Table III., is 62000. Therefore, 
by the rule, 16 x 150 x 5^= 300000. This divided by 
(i X 2^ X 62000 =) 496000 gives 0-6048, the required deflec- 
tion — nearly f of an inch. 

I4-4-. — Deflection of a L,ever : Weight w^hen at End. — By 

a transposition of the factors in equation (37.), we obtain — 

This result is equal to one sixteenth of that shown in equa- 
tion (3i.)j 3, rule for the weight at the middle. Therefore, 
for— 

Rule XXXIII.— Proceed as directed in Rule XXVII. ; 
divide the quotient there obtained by 16, and the resulting 
quotient will be the required weight in pounds. 

Example. — What weight is required at the end of a 4 x 12 
inch Georgia-pine lever 20 feet long to deflect it three 
quarters of an inch? Proceeding by Rule XXVII. , we ob- 
tain a quotient of 3823-2; this divided by 16 gives 238 -95^ . 
say 239, the required weight in pounds. 



DEFLECTION OF LEVERS WITH UNIFORM LOAD. I2I 

145. — Deflection of a Lever : JBreadtli or Depth, Load 
at End. — A transposition of the factors of equation (38.) 
gives — 

bd'^—^^, (39.) 

a rule by which to obtain the sectional area of the lever. 
By comparison with equation (32.) it is seen that the result 
in (39.) is 16 times that found by (32.). Therefore, the dimen- 
sions for a lever loaded at the end may be found by — 

Rule XXXIV.— Multiply by 16 the first quotient found 
by Rule XXVII., and then proceed as farther directed in 
Rule XXVII., using the product of 16 times the quotient, 
instead of the said quotient. 

Example. — What should be the size of a spruce lever 20 
feet long, between weight and wall, to sustain 2000 pounds 
at the end with a deflection of i inch ? Proceeding by Rule 
XXVII., we obtain a first quotient of 4571-43. By Rule 
XXXIV., 4571-43 X 16 = 73144.88. Now, if the depth be 
fixed, say, at 20, then 73144-88 divided by (20 x 20 x 20 =) 
8000 gives 9- 143, the required breadth. But to obtain the 
depth, fixing the breadth, say, at 9, we have for 73144-88 di- 
vided by 9 = 8127-21, the cube root of which is 20- 1055, the 
required depth. Again, if the breadth and depth are to be 
in proportion, say, as 0-7 to i-o, then 73144-88 divided by 
0-7 gives 104492-7, the square root of which is 323-254, of 
which the square root is 17-98, the required depth in inches ; 
and 17-98 X 0-7 =r 12-586, the required breadth in inches. 
The lever, therefore, should be, say, I2|-x 18 inches. 

14-6. — Deflection of Levers : Weight Uniformly Distrib- 
uted. — A comparison of the effects of loads upon levers 
shows {Transverse Strains^ Art. 347) that the deflection by a 
uniformly distributed load is equal to that which would be 
produced by three eighths of that load if suspended from 
the end of the lever. Or, P — \ U. Substituting this value 
of P, in equation (37.)» gives — 

i6x|^;^^ 

which reduces to — 

<^=^^, (40-) 



122 CONSTRUCTION. 

a rule for the deflection of levers loaded Avith an equally dis- 
tributed load. 

147i — Reflection of I^evers witU Uniformly l>l§tributecl 
Ijoad. — The deflection shown in equation (40.) is just six 
times that shown in equation (33.). The result by (33.) mul- 
tiplied by 6 will equal the result by (40.); therefore, we 
have — 

Rule XXXV.— Proceed as directed in Rule XXVIII.; 
the result thereby obtained multiplied by 6 will give the 
required deflection. 

Example. — To what depth will 500 pounds deflect a 3 x 10 
inch white-pine lever 10 feet long, the weight uniformly 
distributed over the lever? Here, by Rule XXVIII., we 
obtain the result 0-05747 ; this multiplied by 6 gives 0-3448, 
the required deflection. 

148. — Deflection of Ijever§ : IVeiglit when Uniformly 
Distributed. — By a transposition of factors in (40.), we ob- 
tain — 

TT Fb d' d / ■ 

This is equal to one sixth that of equation (31.) ; therefore, 
we have — 

Rule XXXVI. -Proceed as directed in Rule XXVI.; 
the quotient thereby obtained divide by 6, and the quotient 
thus obtained will be the required weight. 

Example. — What weight will be required to deflect a 
4x5 inch spruce lever i inch, the weight uniformly dis- 
tributed over its length ? Proceeding as directed in Rule 
XXVI., the result thereby obtained is 1750; this divided by 
6 gives 291 f, the required weight in pounds. 

149. — Deflection of L<evers : Kreadttn or Deptli, Load 
Uniformly Distributed. — A transposition of factors in equa- 
tion (41.) gives — 

^^ ^ ~Ts • (42-) 



SIMPLICITY IN CONSTRUCTION. 1 23 

This result is just six times that of equation (32.); wc, there- 
fore, have — 

Rule XXXVIL— Proceed as directed in Rule XXVII. ; 
multiply the first quotient thereby obtained by 6 ; then in 
the subsequent directions use this multiplied quotient in- 
stead of the said first quotient, to obtain the required breadth 
and depth. 

Example. — What should be the size of a spruce lever 10 
feet long, sustaining 2666| pounds, uniformly distributed 
over its length, with a deflection of i inch ? Proceeding 
by Rule XXVII., the first quotient obtained is 761-905; 
this multiplied by 6 gives 4571-43, the multiplied quotient 
which is to be used in place of the said first quotient. Now, 
to obtain the breadth, the depth being fixed, say, at 10 ; 
4571-43 divided by (cube of 10 =) 1000, the quotient, 4- 57, is 
the required breadth. But if the breadth be fixed, say, at 
4, then, to obtain the depth, 4571-43 divided by 4 gives 
1142-86, the cube root of which is 10-46, the required depth. 
Again, if the breadth and depth are to be in proportion, say, 
as 0-6 to I -o, then 4571 -43 divided by o-6 gives 7619-05, the 
square root of which is 87-27, of which the square root is 
9-343, the required depth in inches ; and 9-343 x o-6 equals 
5-6, the required breadth in inches; or, the lever maybe, 
say, 5f X g^ inches. 

CONSTRUCTION IN GENERAL. 

150. — Construction: Object Clearly Defined. — In the 

various parts of timber construction, known as floors, par- 
titions, roofs, bridges, etc., each has a specific object, and in 
all designs for such constructions this object should be kept 
clearly in view, the various parts being so disposed as to 
serve the design with the least quantity of material. The 
simplest form is the best, not only because it is the most 
economical, but for many other reasons. The great number 
of joints, in a complex design, render the construction liable 
to derangement by multiplied compressions, shrinkage, and, 
in consequence, highly increased oblique strains ; by which 
its stability and durability are greatly lessened. 



124 



CONSTRUCTION. 
FLOORS. 



151. — Floors I>e§cribed. — Floors are most generally con, 
structed siitgle; that is, simply a series of parallel beams, each 




Fig. 39. 



Spanning the width of the building, as seen at Fig. 39. Oc- 




Fig. 40. 

casionally floors are constructed double, as at Fig. 40 ; and 
sometimes framed, as at Fig. 41 ; but these methods are 



RULES APPLIED TO FLOORS. 



25 



seldom practised, inasmuch as either of these requires more 
timber than the single floor. Where lathing and plastering 
is attached to the floor-beams to form a ceiling below, the 
springing of the beams, by customary use, is liable to crack 
the plastering. To obviate this in good dweUings, the double 
and framed floors have been resorted to, but more in former 
times than now, as the cross-furring (a series of narrow strips 
of board or plank nailed transversely to the underside of 




Fig. 41. 

the beams to receive the lathing for the plastering) serves a 
like purpose very nearly as well. 



as- 



(62. — Floor-Beams. — The size of floor-beams can be 
certained by the preceding rules for the stiffness of materials. 
These rules give the required dimensions for the various 
kinds of material in common use. The rules may be some- 
what abridged for ordinary use, if some of the quantities 
represented in the formula be made constant within certain 
limits. For example, if the load per foot superficial upon 
the floor be fixed, and the deflection, then these, together 
with the constant represented by F, may be reduced to one 



126 CONSTRUCTION. 

constant. For dwellings, the load per foot may be taken at 
70 pounds, the weight proper to be allowed for a crowd 
of people on their feet. {Transverse Strains, Art. 114.) To 
this add 20 for the weight of the material of which the floor 
is composed, and the sum, 90, is the value of/, or the weight 
per foot superficial for dwellings. Then <://== U {Art. 130). 
The rate of deflection allowable for this load may be fixed 
at 0-03 inch per foot of the length, or (5^ = 0-03 /. Substitut- 
ing these values in equation (35.), we obtain — 

I -6 Fx 'Oi I ~ 1-6 X '01 F '~ F 
or — 

bd'=l^cl. (43.) 



Putting/ to represent — -p—, we have — 



i875_ 
F 

bd^=jcl\ (44.) 



T R7C 

Now, by reducing — ^ — , for the six woods in common use, 
the value of 7 for each is found as follows: 

Georgia Pine j = 0-32 

Locust J = 0-37 

White Oak / = o-6 

Spruce J = o • 54 

White Pine / = 0-65 

Hemlock / =z c-Sy 

Equation (44.) is a rule for the floor-beams of dwellings ; 
it may be used also to obtain the dimensions of beams for 
stores for all ordinary business • for it will require from 3 to 
5 times the weight used in this rule, or from 200 to 400 
(average 300) pounds to increase the deflection to the limit 
of elasticity in beams of the usual depths and lengths. For 
light stores, therefore, loaded, say, to 150 pounds per foot, 
the beams would be safe, but the deflection would be in- 



CONSTANTS FOR USE IN THE RULES. 12/ 

creased to o-o6 per foot. When so great a deflection as this 
would not be objectionable to the eye, then this rule (44.) 
will serve for the beams of light stores. But for first-class 
stores, taking the rate of deflection at -04 per foot, and 
fixing the weight per superficial foot at 275 pounds, includ- 
ing the weight of the material of which the floor is con- 
structed, and letting k represent the constant, then — 

bd' = kc l\ (45.) 

and for — 

Georgia Pine k = 0-73 

Locust k = 0-85 

White Oak ^ = i .38 

Spruce k = 1-48 

White Pine k = 1-23 

Hemlock k = 1-53 

153. — Floor-Beams for Dwellings. — To find the dimen- 
sions of floor-beams for dwellings, when the rate of deflection 
is 0-03 inch per foot, or for ordinary stores when the load is 
about 150 pounds per foot, and the deflection caused by this 
weight is within the limits of the elasticity of the material, 
we have the following rule : 

Rule XXXVIIL— Multiply the cube of the length by 
the distance apart between the beams (from centres), both in 
feet, and multiply the product by the value of 7* {Art. 152) 
for the material of the beam, and the product will equal the 
product of the breadth into the cube of the depth. Now, 
to find the breadth, divide this product b}^ the cube of the 
depth in inches, and the quotient will be the breadth in 
inches. But if the depth is sought, divide the said product 
by the breadth in inches, and the cube root of the quotient 
will be the depth in inches ; or if the breadth and depth are 
to be in proportion as r is to unity, r representing any re- 
quired decimal, then divide the aforesaid product by the 
value of r, and extract the square root of the quotient, and 
the square root of this square root will be the depth re- 
quired in inches, and the depth multiplied by the value of r 
will be the breadth in inches. 



128 CONSTRUCTION. 

Example, —In a dwelling Or ordinary stOre, what must be 
the breadth of the beams, when placed 15 inches from 
centres, to support a floor covering a span of 16 feet, the 
depth being 1 1 inches, the beams of white oak ? By the 
rule, 4096, the cube of the length, by \\, the distance from 
centres, and by o-6, the value oi j for white oak, equals 
3072. This divided by 1331, the cube of the depth, equals 
2-31 inches, or 2^ inches, the required breadth. But if, in- 
stead of the breadth, the depth be required, the breadth 
being fixed at 3 inches, then the product, 3072, as above, di- 
vided by 3, the breadth, equals 1024 ; the cube root of this 
is io-o8, or, say, 10 inches nearly. But if the breadth and 
depth are to be in proportion, say, as 0-3 to i-o, then the 
aforesaid product, 3072, divided by 0-3, the value of r, 
equals 10240, the square root of which is ioi-2, and the 
square root of this is io-o6, the required depth. This 
multiplied by 0-3, the value of r, equals 3-02, the re- 
quired breadth ; the beam is therefore to be, say, 3 x 10 
inches. 

I54-. — Floor-Beams for First-Class Stores. — To find the 
breadth and depth of the beams for a floor of a first-class 
store sufficient to sustain 250 pounds per foot superficial 
(exclusive of the weight of the material in the floor), with 
a deflection of 0-04 inch per foot of the length, we have — 

Rule XXXIX.— The same as XXXVIII., with the ex- 
ception that the value of /^ {Art. 152) is to be used instead 
of the value of j. 

Example. — The beams of the floor of a first-class store 
are to be of Georgia pine, with a clear bearing between the 
walls of 18 feet, and placed 14 inches from centres: what 
must be the breadth when the depth is 11 inches? By the 
rule, 5832, the cube of the length, and i^, the distance from 
centres, and 0-73, the value of k for Georgia pine, all multi- 
plied together equal 4966-92 ; and this product divided by 
1 33 1, the cube of the depth, equals 3-732, the required 
breadth, or 3f inches. 

But if, instead of the breadth, the depth be required : 
what must be the depth when the breadth is 3 inches? 



DISTANCE APART OF FLOOR-BEAMS. 1 29 

The said product, 4966-92, divided by 3, the breadth, equals 
1655-64, and the cube root of this, 11-83, o^> say, 12 inches, 
is the depth required. 

But if the breadth and depth are to be in a given pro- 
portion, say 0-35 to i-o, the 4966-92 aforesaid divided by 
0-35, the value of r, equals 14191, the square root of 
which is 119-13, and the square root of this square root is 
10-91, or, say, 11 inches, the required depth. And 10-91 
multiplied by 0-35, the value of r, equals 3-82, the required 
breadth, say 3^ inches. 

155, — Floor -Beams: Distance from Centres. — It is 

sometimes desirable, when the breadth and depth of the 
beams are fixed, or when the beams have been sawed and 
are now ready for use, to know the distance from cen- 
tres at which such beams should be placed in order that the 
floor be sufficiently stiff. By a transposition of the factors 
in equation (44.), we obtain — 



jl 



rr.' (46.) 



In like manner, equation (45.) produces — 

These, in words at length, are as follows : 

Rule XL. — Multiply the cube of the depth by the breadth, 
both in inches, and divide the product by the cube of the 
length in feet multiplied by the value of 7', for dwellings 
and for ordinary stores, or by k, for first-class stores, and 
the quotient will be the distance apart from centres in feet. 

Example. — A span of 17 feet, in a dwelling, is to be cov- 
ered by white-pine beams 3x12 inches: at what distance 
apart from centres should they be placed? By the rule, 
1728, the cube of the depth, multiplied by 3, the breadth, 
equals 5184. The cube of 17 is 4913 ; this by 0-65, the value 
of j for white pine, equals 3193-45. The aforesaid 5184 
divided by this 3193-45 equals 1-6233 feet, or, say, 20 inches. 



130 



CONSTRUCTION. 



156. — Framed Openings for Cliimneys and Stairs. — 

Where chimneys, flues, stairs, etc., occur to interrupt the 
bearing, the beams are framed into a piece, b {Fig. 42), called 
a header. The beams, a a, into which the header is framed 
are called trimmers or carriage-beams. These framed beams 
require to be made thicker than the common beams. The 
header must be strong enough to sustain one half of the 
weight that is sustained upon the /^//-beams, c c (the wall at 
the opposite end or another header there sustaining the other 
half), and the trimmers must each sustain one half of the 
weight sustained by the header in addition to the weight it 
supports as a common beam. It is usual in practice to make 




Fig. 42. 

these framed beams one inch thicker than the common beams 
for dwellings, and two inches thicker for heavy stores. This 
practice in ordinary cases answers very well, but in extreme 
cases these dimensions are not proper. Rules applicable 
generally must be deduced from the conditions of the case — 
the load to be sustained and the strength of the material. 

(57. — Breadtli of Headers. — The load sustained by a 
header is equally distributed, and is equal to the superficial 
area of the floor supported by the header multiplied by the 
load on every superficial foot of the floor. This is equal to 
the length of the header multiplied by half the length of the 
tail-beams, and by the load per superficial foot. Putting g 



I 



DIMENSIONS OF HEADERS. I3I 

for the length of the header, n for the length of the tail- 
beams, and / for the load per superficial foot ; U, the uni- 
formly distributed load carried by the header, will equal J 
f n g. By substituting for U, in equation (35.), this value of 
it, we obtain — 

i'6F6 

The symbols g and / here both represent the same thing, 
the length of the header ; combining these, and for d putting 
its value ^r, we obtain— 

1'2 Fr 

To allow for the weakening of the header by the mor- 
tices for the tail-beams (which should be cut as near the 
middle of the depth of the header as practicable), the depth 
should be taken at, say, one inch less than the actual depth. 
With this modification, we obtain — 

i = _^^^3. (48.) 

If /be taken at 90, and r at 0-03, we have, by reducing — 

Fid- if ^^^^ 

which is a rule for the breadth of headers for dwellings and 
for ordinary stores. This, in words, is as follows : 

Rule XLI. — Multiply 937-5 times the length of the tail- 
beams by the cube of the length of the header, both in feet. 
The product divided by the cube of one less than the depth 
multiplied by the value of F, Table III., will equal the 
breadth of the header in inches for dwellings or ordinary 
stores. 

Example. — A header of white pine, for a dwelling, is 10 
feet long, and sustains tail-beams 20 feet long ; its depth is 
12 inches: what must be its breadth? By the rule, 
937-5 X20X 10^=18750000. This divided by (12- r)^x 2900= 



132 CONSTRUCTION. 

3859900, equals 4-858, say 5 inches, the required breadth. 
Y ox first-class stores, f ^\\o\x\^ be taken at 275, and r at 0-04. 
With these values the constants in equation (48.) reduce to 
2148-4375, or, say, 2150. This gives— 

7. 2150 «^3 r . 

a rule for the breadth of a header for first-class stores. It 
is the same as that for dwellings, except that the constant 
2150 is to be used in place of 937-5. Taking the same ex- 
ample, and using the constant 2150 instead of 937-5, we 
obtain 1 1 • 14 as the required breadth of the header for a first- 
class store. Modifying the question by using Georgia pine 
instead of white pine, we obtain 5 -476 as the required thick- 
ness, say 5^ inches. 

158. — Breadth of Carriage-Beams. — A carriage-beam or 
trimmer, in addition to its load as a common beam, carries 
one half of the load on the header, which, as has been 
seen in the last article, is equal to one half of the superficial 
area of the floor supported by the tail-beams multiplied by 
the weight per superficial foot of the load upon the floor ; 
therefore, when the length of the header in feet is repre- 
sented by g, and the length of the tail-beams by n, w equals 

^ X - x/, equals \ f g n.^ 

For a load not at middle, we have (25.) — 

^ '^ = —bT- 

* The load from the header, instead of he'ing ^ f g n, is, more accuratel)", 
ifn{g — c)\ because the surface of floor carried by the header is only 
that which occurs between the surfaces carried by the carriage-beams, each of 
which carries so much of the floor as extends half way to the first tail-beam 

Q 

from it, or the distance - ; therefore, the width of the surface carried equals 

the length of the header less [ 2 x - = J r, or ^ — <r. When, however, it is con- 
sidered that the carriage-beam is liable to receive some weight from a stairs or 
other article in the well-hole, the small additional load above referred to is 
not only not objectionable, but is really quite necessary to be included in the 
calculation. 




THICKNESS OF CARRIAGE-BEAMS. 133 

This is a rule based upon resistance to rupture. By substi- 

/> / 
tuting- for a, the factor of safety, ^ , , its value in terms of 

resistance to flexure {Transverse Strains, (154.)), we have — 

, ,, ^WBlinn 4. Win 71 
a - 



or- 



In this expression, W is a concentrated weight at the dis- 
tances VI and n from the two ends of the beam. Taking the 
load upon a carriage-beam due to the load from the header, 
as above found, and substituting it for JF, we obtain — 

^ ^^3 _ 4 X \fgn m n ^ fgm n ' ^ 
Fr Ft 

This is the expression required for the concentrated load. 
To this is to be added the uniformly distributed load upon 
the carriage-beam ; this is given in equation (35.). Substi- 
tuting for U of this equation its value, f c /, gives — 



i'6F6 . Fr 
Combining these two equations, we have for the total load — 

,^==/k!^i±if.a (51.) 

Fr 

If, in this equation, /be taken at 90, and r at 0-03, these 
reduce to 3000 ; therefore, with this value of -, we have — 

3000{gmn' + %cr) . . 

d = -^rjj . i52.) 

This rule for the breadth of carriage-beams with one 
header, for dwellings and for ordinary stores, is put in words 
as follows : 



134 CONSTRUCTION. 

Rule XLII. — Multiply the length of the framed opening 
by its breadth, and by the square of the length of the tail- 
beams ; to this product add | of the cube of the length into 
the distance of the common beams from centres — all in feet ; 
divide 3000 times the sum by the cube of the depth in inches 
multiplied by the value of /'"for the material of the beam., in 
Table HI., and the quotient will be the breadth in inches. 

Example, — In a tier of 3 x 10 inch beams, placed 14 inches 
from centres, what should be the breadth of a Georgia-pine 
carriage-beam 20 feet long, carrying a header 12 feet long, 
having tail-beams 15 feet long? Here the framed opening 
is 5x12 feet. Therefore, according to the rule, 12x5x15^== 
13500; to which add(|x 20^x-if =) 5833!-; the sum is 19333J, 
and this by 3000= 58000000. The value of F for Geoi-gia 
pine, in Table III., is 5900; the cube of the depth is 1000; 
the product of these two is 5900000 ; therefore, dividing 
the above 58000000 by 5900000 gives a quotient of 9.83, 
the required breadth in inches. If, in equation (51.), / be 

taken at 275, and r at 0-04, then - becomes 6875, and the 
equation becomes — 

b = j-di ' (53-) 

a rule for the breadth of carriage-beams iov first-elass stores ; 
the same as that for dwellings, except that the constant is 
6875 instead of 3000. 

159. — Breadth of Carriage-Beams Carrying Two Sets 
of Tail-Beams. — A rule for this is the same as that for a car- 
riage-beam carrying one set of tail-beams, if to it there be 
added the effect of the second set of tail-beams. Equation 
(51.) with the addition named becomes — 

in which ;/ is the length of one set of tail-beams, and s the 
lensrth of the other set ; and ;;/ + n = I. 



CARRIAGE-BEAMS WITH TWO HEADERS. 1 35 

If / be taken at. 90, and r at 0-03, these two reduee to 
3000, and we have — 

b^- -j^, > 155.) 

a rule for the breadth of a carriage-beam carrying two sets 
of headers, for dwelHngs and for ordinary stores. It may 
be stated in words as follows : 

Rule XLl II.— Multiply the length of the longer set of 
tail-beams by the difference between this length and the 
length of the carriage-beam, and to the product add the 
square of the length of the shorter set of tail-beams ; mul- 
tiply the sum by the length of the longer set of tail-beams, 
and by the length of the header ; to this product add f of 
the product of the cube of the length of the carriage- 
beam into the distance apart from centres of the common 
beams ; multiply this sum by 3000 ; divide this product by 
the product of the cube of the depth in inches into the value 
of F ior the material of the carriage-beam, in Table III., and 
the quotient will be the required breadth. 

Example. — In a tier of 3 x 12 inch beams, placed 14 inches 
from centres, what should be the breadth of a spruce car- 
riage-beam 20 feet long in the clear of the bearings, carry- 
ing two sets of tail-beams, one of them 9 feet long, the 
other 5 feet ; the headers being 15 feet long ? The difference 
between the longer set of tail-beams and the carriage-beam 
is (20 — 9 =) II feet. Therefore, by the rule, 9x11 + 5'' = 
124; then (124x9x15=) i6740 + (f X 20^x|f =) 5833^ = 
22573i; then 22573^x3000 = 67720000. Now the value of 
F for spruce. Table III., is 3500; this by 12^, the cube of 
the depth, equals 6048000 ; by this dividing the aforesaid 
67720000, we obtain a quotient of 11-197, the required 
breadth of the carriage-beam. If, in equation (54.), f be 
taken at 275, and r at 0-04, these reduce to 6875, and we 
obtain — 

^ ^ 6875 \_gn{1nn-Vs'')^-lc P'\^ - ^^^^ 

a rule for the breadth of carriage -beams carrying two sets 



136 CONSTRUCTION. 

of tail-beams, in the floors of first-class stores. This is like 
the rule for dwellings, except that the constant is 6875 in- 
stead of 3000. 

(60. — Breadth of Carriage -Beam wUh Well -Hole at 
Middle. — When the framed opening between the two sets of 
tail-beams occurs at the middle, or when the lengths of the 
two sets of tail-beams are equal, then equation (54.) reduces 
to 

and if /be taken at 90, and r at 0'03, these reduce to 3000, 
and we have — 

, 3000 /(^;/ 2 + 1^^ ^^^^ 



a rule for the breadth of a carriage-beam carrying two sets 
of tail-beams of equal length, in the floor of a dwelling or of 
an ordinary store ; and which in words is as follows : 

Rule XLIV.— Multiply the length of the header by the 
square of the length of the tail-beams, and to the product 
add |- of the product of the square of the length of the car- 
riage-beam by the distance apart from centres of the com- 
mon beams ; multiply the sum by 3000 times the length of 
the carriage-beam ; divide the product by the product of 
the cube of the depth into the value of F for the material of 
the carriage-beam, in Table III., and the quotient will be the 
required breadth. 

Example. — In a tier of 3 x 12 inch beams, placed 12 inches 
from centres, what must be the thickness of a hemlock car- 
riage-beam 20 feet long, carrying two sets of tail-beams, 
each 8 feet long, with headers 10 feet long? By the rule, 
10 X 8' + f X I X 20' = 890 ; 890 X 3000 X 20 = 53400000. Now, 
the value of F, in Table III., for hemlock is 2800 ; this by the 
cube of the depth, 1728, equals 4838400; by this dividing 
the former product, 53400000, and the quotient, 11-0367, is 
the required breadth of the carriage-beam. 



CROSS-BRIDGING. 



37 



If, in equation (57.), /be taken at 275, and rat 0.04, these 
will reduce to 6875, and we shall have — 



b=^ 






(59-) 



a result the same as in equation (58.), except that the 
constant is 6875 instead of 3000. Equation (59.) is a rule for 
the breadth of carriage-beams carrying two sets of tail-beams 
of equal length, in the floor of a iirst-class store. In words 
at length, it is the same as Rule XLIV., except that the con- 
stant 6875 is to be used in place of 3000. 

(61.— Cro§§-Briclgiiis, or Herrings-Bone Bridi^ing.— The 
diagonal struts set between floor-beams, as in Fig. 43, are 
known as cross-bridging, or herring- 
bone bridging. By connecting the 
beams thus at intervals, say, of from 
5 to 8 feet, the stiffness of the floor 
is greatly increased. The absolute 
strength of a tier of beams to resist a 
weight uniformly distributed over 
the Avhole tier is augmented but lit- 
tle by cross-bridging ; but the power 
of any one beam in the tier to re- 
sist a concentrated load upon it, as a heavy article of fur- 
niture or an iron safe, is greatly increased by the cross- 
bridging; for this device, by connecting the loaded beam 
with the adjacent beams on each side, causes these beams to 
assist in carrying the load. To secure the full benefit of the 
diagonal struts, it is very important that the beams be well 
secured from separating laterally, by having strips, such as 
cross-furring, firmly nailed to the under edges of the beams. 
The tie thus made, together with that of the floor-plank on 
the top edges, will prevent the thrust of the struts from sep- 
arating the beams. 

162.— Bridging: Value to Resist Concentrated Lioad§. — 

A rule for determining the additional load which any one 
beam connected by bridging will be capable of sustaining, 
by the assistance derived from the other beams, through the 




Fig. 43. 



138 CONSTRUCTION. 

bridging, may be found in Chapter XVI II., Transverse Strains. 
This rule may be stated thus : 

R = ^^^(i + 2^ + 3 V 4^4- eie) ; (60.) 

in which R is the increased resistance, equal to the addi- 
tional load which may be put upon the loaded beam ; c is the 
distance from centres in feet at which the beams in the tier 
are placed ; /is the load in pounds per superficial foot upon 
the floor ; / is the length of the beams in feet ; and ei is the 
depth of the beams in inches. The squares within the 
bracket are to be extended to as many places as there are 
beams on each side which contribute assistance through the 
bridging. The rule given in the work referred to, for ascer- 
taining the number of spaces between the beams, is — 

or, the depth of the beam in inches divided by the square of 
the distance from centres, in feet, at which the beams are 
placed will give the number of spaces between the beams 
which contribute on each side in sustaining the concentrated 
load. The nearest whole number, minus unity, will equal 
the required number of beams. 

The value of <: for beams in floors of dwellings is given in 
equation (46.), and lor those in first-class stores in equation 
(47.). By a modification of equation (34.), putting e f I for 
U, we have — 

cfl^ r ' 

and— c = y-p , (62.) 

\'6Fbd^r r.. 

or— c =r — (63.) 

These equations give general rules for the value of c. 



INCREASED LOAD BY CROSS-BRIDGING. 1 39 

Now, the rule, in words at length, for the resistance offered 
by the adjoining beams to a weight concentrated upon one 
of the beams sustained by cross-bridging to the others, is — 

Rule XLV". — Divide the depth of the beam in inches by 
the square of the distance apart from centres in feet at 
which the floor-beams are placed ; from the quotient deduct 
unity, and call the whole number nearest to the remainder 
the First Result. Take the sum of the squares of the con- 
secutive numbers from unity to as many places as shall equal 
the above first result ; multiply this sum by 5 times the 
length in feet, by the load per foot superficial upon the floor, 
and by the fifth power of the distance apart from centres in 
feet at Avhich the beams are placed ; divide the product by 
4 times the square of the depth in inches, and the quotient 
will be the weight in pounds required. 

Example. — In a tier of 3 x 12 inch floor-beams 20 feet 
long, placed in a dwelling 16 inches from centres and well 
bridged, what load maybe uniformly distributed upon one of 
the beams, additional to the load which that beam is capable 
of sustaining safely when unassisted by bridging? Here, 
according to the rule, 12 divided by (iJ-4-ii- = ) i-J equals 
6f ; 6|— I = 5f, the nearest whole number to which is 6, the 
first result. The sum of the square of the first 6 numbers 
equals (i + 2' + 3^ + 4'+ 5' + 6' =) i +4 + 9 + 16 + 25 + 36 = 91. 
Therefore, 91 x 5 x 20 x 90 x (|)' ~ 345 1266.'^ The square of 
the depth (12 x 12 = ) 144x4 = 576; by this dividing the 
above 3451266, we have the quotient 5991-78, say 5992 
pounds, the required weight. This is the additiojial load 
which may be placed upon the beam. At 90 pounds per 
superficial foot, the common load on each beam, we have 



* The value of r, 16 inches, equals | feet. The fifth power of this, or (5)*, 
is obtained by involving both numerator and denominator to the fifth power, 
and dividing the fifth power of the former by the fifth power of the latter ; for 

(1)5 = z_. For the numerator we have 4x4x4x4x4=1024, and for the de- 

3^ 
nominator 3x3x3x3x3 = 243. The former divided by the latter gives as a 

quotient 4-214, the value of (|)^ The process of involving a number to a high 

power, or the reverse operation of extracting high roots, may be performed by 

logarithms with great facility. (See Art. 427.) 



I40 CONSTRUCTION. 

90 X 20 X 4. = 2400 as the common load. To this add 5992, 
the load sustained through the bridging by the other beams, 
and the sum, 8392 pounds, will be the total load which may 
be safely sustained, uniformly distributed, upon one beam — 
nearly 3J times the common load. 

163.— Oirclcrs.— When the distance between the walls of 
a building is greater than that which would be the limit for 
the length of ordinary single beams, it becomes requisite to 
introduce one or more additional supports. Where sup- 
ports are needed for a floor and partitions are not desirable, 
it is usual to use a large piece of timber called a girder, sus- 
tained by posts set at intervals of from 8 to 15 feet; or, when 
posts are objectionable, a framed construction called a 
framed girder {Art, 196); or an iron box called a tubular 
iron girder {Art. 182). When a simple timber girder is used 
it is advisable, if it be large, to divide it vertically from end 
to end and reverse the two pieces, exposing the heart of the 
timber to the air in order that it may dry quickly, and also 
to detect decay at the heart. When the halves are bolted 
together, thin slips of wood should be inserted between 
them at the several points at which they are bolted, in order 
to leave sufficient space for the air to circulate freely in the 
space thus formed between them. This tends to prevent 
decay, which \vill be found first at such parts as are not 
exactly tight, nor yet far enough apart to permit the escape 
of moisture. When girders are required for a long bear- 
ing, it is usual to truss them ; that is, to insert between 
the halves two pieces of oak which are inclined towards each 
other, and which meet at the centre of the length of the 
girder like the rafters of a roof-truss, though nearly if not 
quite concealed within the girder. This and many similar 
methods, though extensively practised, are generally worse 
than useless ; since it has been ascertained that, in nearly all 
such cases, the operation has positively ivcakcncd the girder. 

A girder may be strengthened by mechanical contrivance, 
when its depth is required to be greater than any one piece 
of timber will allow. Fig. 44 shows a very simple yet invalu- 
able method of doing this. The two pieces of which the gir- 



CONSTRUCTION OF GIRDERS. I4I 

der is composed are bolted or pinned together, having keys 
inserted between to prevent the pieces from sliding. The 
keys should be of hard wood, well seasoned. The two 
pieces should be about equal in depth, in order that the 
joint between them may be in the neutral line. (See Arts. 
120, 121.) The thickness of the keys should be about half 
their breadth, and the amount of their united thickness 
should be equal to a trifle over the depth and one third of 
the depth of the girder. Instead of bolts orpins, iron hoops 
are sometimes used ; and when they can be procured, they 
are far preferable. In this case, the girder is diminished at 
the ends, and the hoops driven from each end towards the 
middle. A girder may be spliced if timber of a sufficient 
length cannot be obtained ; though not at or near the mid- 




Fig. 44. 

die, if it can be avoided. (See Art. 87.) Girders should 
rest from 9 to 12 inches on each wall, and a space should be 
left for the air to circulate around the ends, that the damp- 
ness may evaporate. 

164-- — Girders: Dimensions. — The size of a girder, for 
any special case, may be determined by equations (21.), (22.), 
(25.), (27.), and (28.), to resist rupture ; and to resist deflection, 
by equations (32.) and (35.)- For girders in dwellings, equa- 
tion (44,) may be used. In this case, the value of c is to be 
taken equal to the width of floor supported by the girder, 
which is equal to the sum of the distances half way to the 
wall or next bearing on each side. When there is but one 



142 CONSTRUCTION. 

girder between the two walls, the value of c is equal to half 
the distance between the walls. The rule for girders for 
dtvellijtgs, in words, is — 

Rule XLVI. — Multiply the cube of the length of the gir- 
der by the sum of the distances from the girder half way to 
the next bearing on each side, and by the value of J for the 
material of the girder, in Art. 152; the product will equal 
the product of the breadth of the girder into the cube of the 
depth. To obtain the breadth, divide this product by the 
cube of the depth ; the quotient will be the breadth. To 
obtain the depth, divide the said product by the breadth ; 
the cube root of the quotient will be the depth. If the 
breadth and depth are to be in a given proportion, say as 
r ', i-o, then divide the aforesaid quotient by the value of 
r ; take the square root of the quotient ; then the square root 
of this square root will be the depth, and the depth multi- 
plied by the value of r will be the breadth. 

Example. — In the floor of a dwelling, what should be the 
size of a Georgia-pine girder 14 feet long between posts, 
placed at 10 feet from one wall and 20 feet from the other? 
The value of c here is J^o, 4. ^0. _ 3_o. — j^^ -phe value oi j 
for Georgia pine {Art. 152) is 0-32. By the rule, 14^ x 15 x 
0-32 = 13171 .2. Now, to find the breadth when the depth 
is 12 inches ; 13171 -2 divided by the cube of 12, or by 1728, 
gives a quotient of 7-622, or 7f, the required breadth. 
Again, to find the depth, when the breadth is 8 inches : 
13171-2 divided by 8 gives 1646-4, the cube root of which is 
II -808, or, say, \\\ inches, the required depth. But if 
neither breadth nor depth have been previously determined, 
except as to their proportion, say as 0-7 to i-o, then 
13171-2 divided by 0-7 gives 18816, of which the square 
root is 137- 171, and of this the square root is 11 -712, or, say, 
I if inches, the required depth. For the breadth, we have 
1 1 -712 by 0-7 equals 8-198, or, say, 8 J, the required 
breadth. Thus the girder is required to be 7f x 12, 8 x iif, 
or8ixiif inches. This example is one in a dwelling or 
ordinary store ; ior first-class stores the rule for girders is the 
same as the last, except that the value of k is to be taken 
instead of/, in Art. 152. 



FIRE-PROOF TIMBER FLOORS. I43 

165. — Solid Timber Floors. — Floors constructed with 
rolled-iron beams and brick arches are proof against fire 
only to a limited degree ; for experience has shown that the 
heat, in an extensive conflagration, is sufficiently intense to 
deprive the iron of its rigidity, and consequently of its 
strength. Singular as it may seem, it is nevertheless true 
that wood, under certain circumstances, has a greater fire- 
resisting quality than iron. Floors of timber constructed, 
as is usual, with the beams set apart, have but little power 
to resist fire, but if the spaces between the beams be filled 
up solid with other beams, which thus close the openings 
against the passage of the flames, and the under surface be 
coated with plastering mortar containing a large portion ot 
plaster of Paris, and finished smooth, then this w^ooden 
floor will resist the action of fire longer than a floor of iron 
beams and brick arches. The wooden beams should be se- 
cured to each other by dowels or spikes. 

166. — Solid Timber Floors for Dwellings and Assem- 
bly-Rooms. — From Transverse Strains, Art. 702, we have — 

(82+j^/V^ 
0-576/^ ' 

which may be modified so as to take this form : 

0-576^^ ^ ^^ 

which is a rule for the depth or thickness of solid timber 
floors for dwellings, assembly-rooms, or office buildings, and 
in which y and // are constants depending upon the mate- 
rial ; thus, for — 

Georgia Pine jF = 4, and // = 0-3x4 

Spruce / = 2-5", " // = 0-365 

White Pine y = 2 J, " /i = 0-389 

Hemlock J = 2, " // = o-39 

The rule may be stated in words thus : 

Ru/e XLVII. — Multiply the length by the value of y, 



144 CONSTRUCTION. 

and by the value of //, as above given ; to the product add 
82 ; multiply the sum by the cube of the length ; divide this 
product by 0.576 times the value of F, in Table III. ; then 
the cube root of the quotient will be the required depth in 
inches. 

^^^;;///<f.— What depth is required for a solid Georgia- 
pine floor to cover a span of 20 feet ? For Georgia pine 
i^=5900; y, as above given, equals 4, and h equals 0-314; 
therefore, by the rule — 

^_ | /(824-4 X Q- 3 ^^W^^' ^ 1/856965 ^ ^ o . 
0.576x5900 3398-4 ' 

or, the depth required is, say, 6.32 or 6j^^ inches. 

167. — Solid Timber Fioor§ for First-Class Stores. — The 

equation given for first-class stores, in Transverse Strains^ 
Art. 702, is — 

^3_ {261 -V yd) P 

which may be changed to this form : 

in which y is as before, and k for — 

Georgia Pine equals 0-4 

Spruce equals 0.472 

White Pine equals o. 502 

Hemlock equals 0-506 

This rule may be put in words the same as Rule XLVII., 
except as to the constants, which require that 263 be used in 
place of 82, that k be used in place of //, and that 0.768 be 
used in place of 0-576. Table XXI. of Transverse Strains 
contains the results of computation showing the depths of 
solid timber floors for dwellings and assembly-rooms and 
for first-class stores, in floors of spans varying from 8 to 30 
feet, and for the four kinds of timber before named. 



IRON FLOOR-BEAMS. 



145 



168, — Rolled - Iron Beam§. — The dimensions of iron 
beams, whether wrought or cast, are to be ascertained by 
the rules already given, when the beams are of rectangular 
form in their cross-section ; these rules are applicable alike 
to wood and iron {ArL 93), and may be used for any mate- 
rial, provided the constant appropriate to the given mate- 
rial be used. But when the form of cross- 
section is such as that which is usual for 
rolled-iron beams (Fzo-. 45), the rules 
need modifying. Without attempting 
to explain these modifications (referring 
for this to Transverse Strains, Art. 457 
and following article), it may be re- 
marked that the elements of resistance 
to flexure in a beam constitute what is 
termed the Moment of Inertia. This, in 
a beam of rectangular cross-section, is 
equal to y^ o^ the breadth into the cube of the depth ; or- 




FiG. 45. 



/ 



i^bd\ 



(66.) 



This would be appropriate to rolled-iron beams if the 
hollow on each side were filled with metal, so as to complete 
the form of cross-section into a rectangle. The proper ex- 
pression for them may be obtained by taking first the 
moment for the beam as if it were a solid rectangle, and 
from this deducting the moment for the part which on each 
side is wanting, or for the rectangles of the hollows. In 
accordance with this view of the case, we have — 



/=yV(^^^-^.<^); {67.) 

in which d is the breadth of the beam or width of the 
flanges ; d^ is the breadth of the two hollows, or is equal to/5 
less the thickness of the zued or ste7n ; d is the depth includ- 
ing top and bottom flanges ; and d^ is the depth in the clear 
between the top and bottom flanges. 

Now, if equation (32.) be divided by 12, we shall have — 



12 



12FJ' 



146 CONSTRUCTION. 

and since -^^ ^ ^^ represents the moment of inertia, weViave- 



1 = 



12 F 6 



(68.) 



This gives the vakie of / for a beam of any form in cross- 
section loaded at the middle. By this equation the values 
of / have been computed for rolled -iron beams of many 
sizes, and the results recorded in Table XVII., Transverse 
Strahis. A few of these are included in Table IV., as follows : 



Table IV. — Rolled- Iron Beams. 



Name. 



Trenton.. 
Paterson. 
Phoenix. , 
Trenton . 
Phoenix . 
Trenton . 
Buffalo.., 
Paterson. 
Phoenix .. 
Phoenix . 



Depth. 



Weight 
per yard. 



30 
30 
36 
40 

55 
60 

65 
80 
70 
84 



7-84 
12-082 
14-317 
23-761 
42-43 
46-012 
64-526 

84-735 
92-207 

107-793 



Name. 



Buffalo.... 
Phoenix. . 
Buffalo. .. 
Buffalo.... 
Trenton. . 
Buffalo . . . 
Paterson. . 
Paterson.. 
Buffalo.... 
Trenton. . . 



Depth. 


Weight 
per yard. 


9 


90 


9 


150 


loi 


90 


10^ 


105 


loi 


135 


I2i 


125 


I2i 


125 


I2i 


170 


I2i 


180 


ISA 


150 



109-117 

190-63 

151-436 

175-645 

241-478 

286-019 
292-05 

398-936 
418-945 
528-223 



169. — Rolled - Iron 1Seain§: Dimensions; Weight at 
Middle. — If, in equation (68.), there be substituted for F its 
value for wrought iron, as in Table III., we shall have — 



/ 



WP 



1 2 X 62000 6 ' 



or- 



WV 



744000 c> 



(69.) 



This is a rule by which to ascertain the size of a rolled-iron 
beam to sustain a given weight at middle with a given de- 
flection, and, in words at length, is as follows: 

Rule XLVIII. — Multiply the weight in pounds by the 
cube of the length in feet ; divide the product by 744000 
times the deflection in inches, and the quotient will be the 



DEFLECTION OF IRON BEAMS. I47 

moment of inertia of the required beam, and may be found, 
or the next nearest number, in Table IV. in column headed 
/. Opposite to the number thus found, to the left, will be 
found the name, depth, and weight per yard of the required 
beam. 

Example. — Which of the beams of Table IV. would be 
proper to carry 10,000 pounds at the middle with a deflection 
of one inch, the length between bearings being 20 feet ? 
Here we have, substituting for the symbols their values — 

W l^ 10000x20^ 80000000 
7440DO 6 744000 X I 744000 / :j / ' 

or, the momentof inertia of the required beam is 107-527, the 
nearest to which, in the table, is 107-793, pertaining to the 
Phoenix 9-inch, 84-pound beam. This, then, is the required 
beam. 

170. — Rolled-Iron Beams: Deflection ^vhen IVeiglit is 
at Hiddle. — By a transposition of symbols in equation (69.), 
we have — 

_ Wl^ 

~ 744000 /' (70-) 

or a rule for the deflection of rolled-iron beams when the 
weight is at the middle. This, in words, is — 

Ride XLIX. — Multiply the weight in pounds b}- the 
cube of the length in feet ; divide the product by 744000 
times the value of / for the given beam, and the quotient 
will be the required deflection in inches. 

Example. — What will be the deflection of a Phoenix 9- 
inch, 70-pound beam 20 feet long, loaded at the middle with 
7500 pounds? The value of / for this beam, in Table IV., 
is 92 • 207 ; therefore, substituting for the symbols their val- 
ues, and proceeding by the rule, we have — 

W P 7500x20^ _ ^ 

d = = • — — = 0.87461 ; 

744000 / 744000 x 92 • 207 

or, the deflection will be, say, f of an inch. 



148 CONSTRUCTION. 

171. — Rolled -Iron Beams: Tl^eiglit when at middle.— 

A transposition of factors in equation (70.) gives — 

_ 744000 / S 

P ' (71.) 

This is a rule for the weight at middle, and, in words, is — 
Ru/e L. — Multiply 744000 times the value of / by the 

deflection in inches ; divide the product by the cube of the 

length, and the quotient will be the required weight in 

pounds. 

Example. — What weight at the middle of a Buffalo 9-inch, 

90-pound beam will deflect it one inch, the length between 

bearings being 20 feet ? The value of / for this beam, in 

Table IV., is 109- 117; therefore — 

744000/ (J 74400QXI09-II7 XI ,^^^„ p. 
P 20^ ^/ ' 

or, the required weight is, say, 10,148 pounds. 

(72. — Rolled -Iron Beam§ : Weiglit at any Point. — The 

equation for a load at any point is ( Transverse Strains, Art. 
485)- 

186000 IS 
~ I in n ' (72.) 

in which wand n represent the two parts in feet into which 
the point where the load rests divides the length. This, in 
words, is as follows : 

Rule LI. — Multiply 186000 times the value of / by the 
deflection in inches ; divide the product b}^ the product of 
the length into the rectangle formed by the two parts into 
which the point where the load rests divides the length ; 
the quotient will be the required weight in pounds. 

Example. — What weight is required, located at 10 feet 
from one end, to deflect i^ inches a Paterson 12^-inch, 125- 
pound beam 25 feet long between bearings ? The value of 
/for this beam, in Table IV., is 292-05 ; /;/ = 10, and n — I 
— 7;/ = 25 — 10 -^ 15 ; therefore — 



VARYING WEIGHTS ON IRON BEAMS. I49 

i86ooo/(^ 186000x292-05x1.5 

[[/ = ■ = • ^--^ = 21728-52; 

/ 7n u 25 X 10 X 15 ' -^ ' 

or, the required weight is, say, 21,730 pounds. 

173,— Rolled-Iron Beams: Dimensions; Weig^ht at any 
Point.— By transposition of factors in equation (72.), we ob- 
tain — 

_ W I m n 
^ ~ 186000^- (73-) 

This may be expressed in words as follows : 

Rule LII. — Multiply the weight by the length, and by the 
rectangle of the two parts into which the point where the 
weight rests divides the length ; divide the product by 186000 
times the deflection, and the quotient will be the value of /, 
which (or its next nearest number) may be found in Table 
IV., opposite to which will be found the required beam. 

Example. — What beam 10 feet long will be required to 
carry 5000 pounds at 3 feet from one end Avith a deflection 
of 0-4 inch? Here we have m equal 3, and 71 equal 7; 
therefore — 

_ W I m n 5000 X 10x3x7 

~ 1860006' ~ 186000x0-4 ~ 4* i- 

The value of / is 14- 113, the nearest number to which in 
the table, is 14-317, the moment of inertia of the Phoenix 5- 
inch, 36-pound beam ; this, therefore, is the beam required. 

174. — Rolled-Iron Beams: Dimensions; Weiglit Uniform- 
ly Distributed. — Since %U — W{Art. 138), equation (69.) may 
be modified by the substitution of this value of W^ when we 
obtain — 



/ 
which reduces to — 



JJLIl 

744000 s' 



1 160400 o 



I50 CONSTRUCTION. 

a rule for the dimensions of a beam for a uniformly distrib- 
uted load, which, in words, is as follows : 

Rule LI 1 1. — Multiply the uniformly distributed load by 
the cube of the length ; divide the product by 1 190400 times 
the deflection, and the quotient will be the value of /, corre- 
sponding to which, or to its next nearest number will be 
found in Table IV. the required beam. 

Example. — What beam 10 feet long is required to sus- 
tain an equally distributed load of 14,000 pounds with a de- 
flection of half an inch ? For this we have — 

14000 X 10^ 

I = -—^ : = 23 • 52. 

1190400x0-5 ^ -^ 

This is the moment of inertia of the required beam ; nearly 
the same as 23-761, in Table IV., the value of / for a Tren- 
ton 6-inch, 40-pound beam, which will serve as the re- 
quired beam. 

175. — Rolled-Iron Beams : I>efleeUoii ; WeiglitUiiiforinJy 

Distributed. — A transposition of the factors in equation (74.) 
gives — 

1 190400/' (75-) 

a rule for the deflection of a uniformly loaded beam, and 
which may be put in these words, namely : 

Rule LIV. — Multiply the uniformly distributed load by 
the cube of the length ; divide the product by 1 190400 times 
the value of /, Table* IV., and the quotient will be the re- 
quired deflection. 

Example. — To what depth will 14,000 pounds, uniformly 
distributed, deflect a Buffalo io|^-inch, 90-pound beam 20 
feet long ? The value of / for this beam, as per the table, is 
151-436; therefore — 

14000 X 20^ ^ 

d — -pr — 0-6213 ; 

1 190400 X 151 -436 

or, the required deflection is, say, f of an inch. 



IRON FLOOR-BEAMS FOR DWELLINGS. 151 

176. — Rolled -Iron Beams: ^Velght ^irhen Uniforitily 
distributed. — Equation (75.), by a transposition of factors, 

gives — 

£/= "904og _^g, (76.) 

a rule for the weight uniformly distributed, and which may 
be worded thus : 

Rule LV. — Multiply 1190400 times the value of /, Table 
IV., by the deflection ; divide the product by the cube of 
the length, and the quotient will be the required weight. 

Example. — What weight uniformly distributed upon a 
Buffalo loj-inch, 105 -pound beam 25 feet long between 
bearings will deflect it f of an inch ? 

The value of / for this beam, as per Table IV., is 175-645 ; 
therefore — 

_. 1190400 X 175-645 X f . 

or, the required weight is, say, 10,036 pounds. 

(77. — Rolled-Iron Reams: Floors of Dwellings or A§- 
sembly- Rooms. — From Transverse Strains^ Art. 500, we 
have — 

'- P 420' ^^^^ 

a rule for the distance from centres of rolled-iron beams in 
floors of dwellings, assembly-rooms, or offices, where the 
spaces between the beams are hlled in with brick arches and 
concrete. In the equation, c is the distance apart from cen- 
tres in feet, and y is the weight per yard of the beam. This, 
in words, is thus expressed : 

Ride LVI. — Divide 255 times the value of / by the cube 
of the length ; from the quotient deduct one 420th part of 
the weight of the beam per yard, and the remainder will be 
the required distance apart from centres. 

Example. — What should be the distance apart from cen- 



152 CONSTRUCTION. 

tres of Buffalo i2:}-inch, 125 -pound beams 25 feet long 
between bearings, in the floor of an assembly-room? For 
these beams, in Table IV., / equals 286.019, and y— 125; 
therefore — 

_ 255 X 286-019 125 ^ 
~ 25' 420' 

72934-8 125 ^^o 

15625 420 y H- 0/ . 

or, the required distance from centres is, say, 4 feet 4^^ 
inches. 

(78, — Rolled-Iron Beam§: Floors of First-€la§s Storc§. 

— From Transverse Strains, Art. 504, we have — 

148-8/ / 

a rule for the distance from centres of rolled-iron beams in 
the floor of a first-class store ; the spaces between the beams 
being filled with brick arches and concrete. This rule may 
be put in words as follows : 

Rule\N\\. — Divide 148-8 times the value of / by the 
cube of the length ; from the quotient deduct one 960th 
part of the weight of the beam per yard, and the remainder 
will be the distance apart of the beams from centres in 
feet. 

Example. — W]i?± should be the distance apart from cen- 
tres of Buffalo 12^-inch, 180-pound beams 20 feet long 
between bearings, in the floor of a first-class store? For 
these beams the value of /, Table IV., is 418-945, and the 
value of J/ is 180; therefore — 

148-8 X 418-945 180 ^ 

20^ 960 

or, the required distance from centres is, say, 7 feet 7J 
inches. 



TIE-RODS FOR IRON BEAMS. 1 53 

(79. — Floor-Arches: Oeneral Considerations. — In fill- 
ing the spaces between the iron beams of a floor, the arches 
should be constructed with hard whole brick of good shape, 
laid upon the supporting centre in contact with each other, 
and the joints thoroughly filled with cement grout, and 
keyed with slate. Made in this manner, the arches need not 
be over four inches thick at the crown for spans extending 
to 7 or 8 feet, and 8 inches thick at the springing, where 
they should be started upon a proper skew-back. The rise 
of the arch should not be less than i^ inches for each foot 
of the span. 

(80. — Floor - Arches ; Tie -Rods: Dxvcllings. — From 
Transverse Strains, Art. 507, we have — 



<^=i 4/0.0198 <: i", (79.) 

which is a rule for the diameter in inches of a tie-rod for 
an arch in the floor of a bank, office building, or assembly- 
room ; in which d is the diameter in inches of the rod, s is 
the span of the arch, and c is the distance apart between the 
rods (s and c both in feet). This rule requires that the arch 
rise \\ inches per foot of the span, and that the brick-work 
and the superimposed load each weigh 70 pounds, or to- 
gether 140 pounds. This rule, in words, is as follows: 

Rule LVIII. — Multiply the span of the arch by the dis- 
tance apart at which the rods are placed, and by the decimal 
0-0198 ; the square root of the product will be the diameter 
of the required rod. 

Example. — What should be the diameter of the wrought- 
iron ties of brick arches of 5 feet span, in a bank or hall of 
assembly, where the ties are 8 feet apart ? For this we 
have — 

d = ^0-0198 X 8 X 5 = i^-792 = 0-89 ; 

or, the diameter of the required rods should be, say, | of an 
inch. 

(8(. — Floor-Arches; Tie-Rods: First-Class IStores. — From 
the same source as in last article, we have — 



^ = 4/0 • 045 27 c s, (80.) 



154 



CONSTRUCTION. 



which is a rule for the size of tie-rods for the brick arches 
of the floors of first-class stores, where the arches have a 
rise of i|- inches for each foot of the span, and where the 
weight of the brick arch and concrete* is not over 70 pounds 
per superficial foot of the floor, and the loading does not 
exceed 250 pounds per superficial foot. As the rule is the 
same as the one in the preceding article, except the deci- 
mal, a recital of the rule, in words, is not here needed. To 
obtain the required diameter, proceed as directed in Rule 
LVIIL, using the decimal 0-04527 instead of the one there 
given. 

TUBULAR IRON GIRDERS. 

(82. — TiBbiilar Iron Girders: B>e§criptioii. — The use of 

wooden beams for floors is limited to spans of about 25 feet. 
When greater spans than this are to be covered, some expe- 
dient must be resorted to by which 
intermediate bearings for the floor- 
beams may be provided. Wooden 
girders may be used, but these 
need to be supported b}^ posts at 
intervals of from 10 to 15 feet, 
unless the girders are trussed, or 
made up of top and bottom chords, 
struts, and ties. And even this 
is objectionable, owing to the 
height such a piece of framing 
requires, and which encumbers 
the otherwise free space of the 
hall. A substitute for the framed 
girder has been found in the 
tubular iron girder, as in Fig. 46, made of rolled plate 
iron and angle irons, riveted. They require to be stiffened 
by an occasional upright T iron along eachside, and a 
cross-head at least at each bearing. 

183. — Tubular Iron GSrders: Area of Flanges ; Load 
at Middle. — In wrought-iron tubular girders it is usual to 
make the top and bottom flanges of equal thickness. From 
Transverse Strains, Art. 551, we have — 




Fig. 46. 



TUBULAR IRON GIRDERS. 155 

a rule for the area of the bottom flange ; in which a' equals 
the area of the flange in inches, W the weight in pounds at 
the middle, / the length and d the depth of the girder, both 
in feet, and k the safe load in pounds per inch with which 
the metal may be loaded, and which is usually taken at 
9000. The rule may be stated thus : 

Riile LIX. — Multiply the weight by the length : divide 
the product by 4 times the depth into the value of k, and 
the quotient will be the required area of the bottom flange. 

Example. — In a girder 40 feet long and 3 feet high, to 
carry 75,000 pounds at the middle, what area of metal is 
required in the bottom flange, putting k at 9000 ? For this 
we have, by the rule — 

A,d k 4 X 3 X 9000 

or, the area required is 27-I inches. This is the amount of 
uncut metal. An allowance is required for that which will 
be cut by rivet-holes. This is usually an addition of one 
sixth. 

184. — Tubular Iron Oirders : Area of Flanges; Load at 
aiiy Point.— The equation suitable for this {Transverse 
Strains^ Art. 553) is — 

a'-w-^^^' (82.) 

^ -'^ dkl ' ^ ^ 

in which in and n are the distances respectively from the lo- 
cation of the load to the two ends of the girder. The other 
symbols are the same as in the last article. This rule may 
be thus stated : 

Rule LX. — Multiply the weight by the values of m and 
of n\ divide the product by the product of the depth into 
the length and into the value of k, and the quotient will be 
the required area of the bottom flange. 

Example. — In a girder 50 feet long between bearings and 



156 CONSTRUCTION. 

3|- feet high, what area of metal is required in the bottom 
flange to sustain 50,000 pounds at 20 feet from one end, when 
k equals 9000^ By the rule, we have — 

,^^ 171 n 50000 X 20 X xo 

d k I 3^ X 9000 X 50 

or, each flange requires 19 inches of solid metal uncut for 
rivets. 

185. — Tubular Iron Crirders : Area of Flanges; Load 
Uniformly I>i§trlbuted. — The equation appropriate here is 
{Trafisverse Strains^ Art. 555) — 

This is a rule by which to obtain the area of cross-sec- 
tion of the bottom flange at any point in the length of the 
girder, the load uniformly distributed ; in and n being the 
respective distances from the point measured to the two 
ends of the girder, and U representing the uniformly dis- 
tributed load in pounds. This, in words, is described as 
follows : 

Rtile LXI. — Divide the weight by the product of twice 
the depth into the length and into the value of k ; then the 
quotient multiplied by the values of in and of n will be the 
required area of the bottom flange at the point measured, 
the distance of which from the ends equals ;;/ and ;/. 

Example. — In a girder 50 feet long and ih feet high, to 
carry a uniformly distributed load of 120,000 pounds, what 
area of cross-section is required in the bottom flange, at the 
middle and at intervals of 5 feet thence, to each support ; k 
being taken at 9000? Here we have, first — 

^^ in n 120000 mil 

a — U — T-r-r = ' :. " — 0-038095 in n. 

2dkl 2 X 3|- X 9000 X 50 ^ ^^ 

Now, when m — u— 25, we have the middle point ; then — 

a' = 0-038095 m n = 0-038095 x 25 x 25 = 23-81 ; 



SHEARING STRAIN. 15/ 

or, the area of the bottom flange at mid-length is 23-81 
inches. 

When in — 20, then n — 30, and — 

a' = 0- 038095 X 20 X 30 = 22 • 86 ; 

or, the required area, at 5 feet either way from the middle, 
is 22^ inches. 

When 7/1= 15, then 71 = 35, and — 

a = 0-038095 X 15 X 35 = 20-0 ; 

or, at 10 feet either way from the middle, the required area 
is 20 inches. 

When 771 = 10, then 7t =40, and — 

a = 0-038095 X 10x40 = 15-24 ; 

or, at 15 feet either way from the middle, the required area 
is I5:|- inches. 

When 77i = 5, then 71 = 4^, and — 

^' = 0-038095 X 5x45 r= 8-57; 

or, at 20 feet each side of the middle, the required area is 8f 
inches. 

The area of cross-section found in every case is that of 
the uncut fibres ; to this is to be added as much as will be 
cut by the rivets. This is usually about one sixth of the area 
given by the rule. The top flange is to be made equal in 
area to the bottom flange. The flanges are unvarying in 
Avidth from end to end, the variation of area being obtained 
by varying the thickness of the flanges, and this being at- 
tained by building the flange in lamina, or plates ; but these 
should not be less than a quarter of an inch thick. There 
should be added to the length of the girder, in the clear, 
about one tenth of its length for supports on the walls : thus, 
a girder 30 feet long requires 3 feet added for supports, or 
18 inches on each wall. 

186.— Tutoular Iron Girders: Shearing Strain,— The top 

and bottom flanges are provided of sufficient size to i-esist 



158 CONSTRUCTION. 

the transverse strain ; the two upright plates, technically 
termed the web, need, therefore, to be thick enough to resist 
only the shearing strain. This, upon a beam uniformly 
loaded, is at the middle theoretically nothing, but from 
thence it increases regularly towards eac«h support, where it 
equals half the whole weight. For example, the girder of 
Art. 185, 50 feet long between supports, carries 120,000 
pounds uniformly distributed over its length. In this case 
the shearing strain at the wall at each end is the half of 
120,000 pounds, or 60,000 pounds; at 5 feet from the wall it 
is -^^ or \ less, or 48,000 pounds ; at 10 feet from the wall it 
is f less, or 36,000 pounds ; at 1 5 feet it is 24,000 ; at 20 feet 
it is 12,000; and at 25 feet or the middle, it is nothing. 

187- — Tubular Iron Oirder§: Thickness of Web.— The 
equation appropriate for this is — 

'=^'-' (^^-^ 

in which t is the thickness of the web (equal to the sum of 
the thicknesses of the two side plates), d is the height of the 
plate {t and d both in inches), G is' the shearing strain, and k' 
is the effective resistance of wrought iron to shearing per 
inch of cross-section. This may be put in words as follows • 

Ride LXII. — Divide the shearing strain by the product of 
the depth in inches into the value of k' , and the quotient 
will be the thickness of the web, or of the two side plates 
taken together. 

Example, — What is the required thickness of web in a 
girder 50 feet between bearings, side plates 38 inches high 
between top and bottom flanges, and to carry 120,000 pounds, 
uniformly distributed ? Here, putting the shearing resistance 
of the plates at 7000 pounds per inch, we have — 



/. = 



dk 38x7000 266000 



The shearing strain at the supports, as in last article, is 
60000 ; therefore, we have for this point — 



LIGHT IRON GIRDERS. 1 59 



60000 

When G = 48000, then 



t = —^ = 0-225. 

266000 



48000 _ 

/ — -^ = o-i8; 

266000 

and when G = 36000, then — 

36000 

/ = -^p- — 0-135. 

266000 ^^ 

Those nearer the middle of the girder are still less than 
these ; and these are all below the practicable thickness, 
which is half an inch for the two plates. The plates ought 
not in practice ever to be made less than a quarter of an 
inch thick. 

188. — Tubular Iron Girders, for FIoor§ of Di¥ellin§^§, 

Assembly-Rooms, and Offlee Buildings.— When the floors of 
these buildings are constructed with rolled-iron beams and 
brick arches, then the following (Art. 568, Transverse Strains) 
is the appropriate equation for the area of cross-section of 
the bottom flange of the girder : 

, .'=(,4o+^)^_-^x-^^; (85.) 

in which a' is in inches, and c, c\ d, /, ?;/, and n are in feet. 
Also, a' is the area required ; y is the weight per yard of the 
rolled-iron beam of the fioor ; c, their distances from centres ; 
c\ the distance from centres at which the tubular girders are 
placed, or the breadth of floor carried by one girder ; d, the 
depth of the girder; k, the effective resistance of the metal 
per inch in the fianges of the girder ; and m and n are the 
distances respectively from the two ends of the girder to the 
point at which the area of cross-section of the bottom flange 
is required. The rule ma}^ be thus described : 

Ride LXIII. — Divide the weight per yard of the rolled- 
iron beams by 3 times their distance from centres; to the 
quotient add 140 and reserve the sum ; deduct the length 
in feet from 700, and with the remainder as a divisor divide 
700 ; multiply the quotient by the above reserved sum, and 



l6o ' CONSTRUCTION. 

by the value of c' ; divide the product by the product of 
twice the depth into the value of k, and the quotient multi- 
plied by the values of m and of n will be the required area 
of cross-section of the bottom flange at the point in the 
length distant from the two ends equal to 7n and n respec- 
tively. 

Example. — In a floor of 9-inch, 70-pound beams, 4 feet 
from centres, what ought to be the area of the bottom flange 
of a tubular girder 40 feet long between bearings, 2 feet 8 
inches deep, and placed 17 feet from the walls or from other 
girders ; the area of the flange to be ascertained at every 5 
feet of the length ; the value of k to be put at 9000? Here 
y =70, c — 4, c' — ly, I— 40, and d^ 2%, Therefore, by 
the rule — 

/ ( 70 \ 700 17 
a =i\ 140 + I X 5 X m n • 

V ^ 3x4/ 700 - 40 2 X 2f X 9000 ' 

a' — 145 -Sjx I -0606 X 0-0003 54if X ;;2;2; 
. a' = 0-05478 V1 11. 

The values of ;;/ and n are — 

At the middle ;;2 = 20 ; ;/ = 20 

5 feet from middle 7/2 = 15; ;/ = 25 

10 '' '^ " ;;2=io; ;/ = 30 

15 '' " " m— 5 ; ;^ = 35 

These give — 

At the middle ^' = 0-05478 x 20 x 20 = 21 -91 

^' 5 feet from middle... .a' = 0-05478 x 15 x 25 = 20- 54 
"10 " " '' ....^' = 0-05478x10x30=16-43 

"15 '' '' - ...../ = 0-05478 X 5x35= 9-59 

These are the areas of uncut fibres at the points named, in 
the lower flange ; the upper flange requires the same sizes. 

189. — Tubular Iron Oirder§, for Floors of First-Class 

Stores.— The equation proper for this is {Transverse Strains, 
Art. 570)— 



CAST IRON COMPARED WITH WROUGHT. 



l6l 



a' ^ ('320 + -^') - 



700 



c in n 



(86.) 



a rule the same in form as that of the previous article ; hence 
it needs no particular exemplification. 

Rule LXIII. of last article may be used for this case, 
simply by using the constant 320 in place of that of 149. 

CAST-IRON GIRDERS. 

(90. — Cast-iron Girders: Inferior. — Rolled-iron beams 
have been so extensively introduced within a few years as 
to have superseded almost entirely the form^erly much used 
cast-iron beam or girder. The tensile strength of cast iron 
is far inferior to that of wrought iron. This inferiority and 
the contingencies to which the metal is subject in casting 
render it very untrustworthy ; it should not be used where 
rolled-iron beams can be procured. A very substantial gir- 
der to carry a brick wall is made by placing two or more 
rolled-iron beams side by side, and securing them together 
by bolts at mid-height of the web ; placing thimbles or sep- 
arators at each bolt. As there may be cases, however, in 
which cast-iron girders will be used, a few rules for them 
will here be given. 

191.— Cast-iron Girder: Load at Middle. — The form of 
cross-section given to this girder usually is as shown in 
Fig. 47. 

In the cross-section, the bottom flange 
is made to contain in area four times as 
much as the top flange. The strength 
will be in proportion to the area of the 
bottom flange, and to the height or 
depth of the girder at middle. Hence, 
to obtain the greater strength from a 
given amount of material, it is requisite 
to make the upright part, or the web, 
rather thin ; yet, in order to prevent 
injurious strains in the casting while it 
is cooling, the parts should be nearly 
equal in thickness. The thickness of the three parts — web, 




l62 CONSTRUCTION. 

top flange, and bottom flange — may be made in proportion 
as 5, 6, and 8. 

For a weight at middle, the form of the web should be 
that of a triangle ; the top flange forming two straight lines 
declining from the centre each way to the bottom flange at 
the ends, like the rafters of a roof to its tie-beam. From 
Transverse Strains, Art. 583, we have — 



W a l 
4850 <ri^' 



(87.) 



which is a rule for the area in inches of the bottom flange, 
for a load at middle ; the area of the top flange is to be equal 
to one fourth of that of the bottom flange. To secure this, 
make the width of the top flange equal to one third of the 
width of the bottom flange ; the thickness of the former, 
as before directed, being made equal to f or f of the lat- 
ter. The weight W is in pounds ; the length / is in feet ; 
and the depth d is in inches. The factor of safety a should 
be taken at not less than 3 ; better at 4 or 5. 

The equation in words may be as follows : 

Rnle LXIV. — Multiply the weight by the length, and by 
the factor of safety ; divide the product by 4850 times the 
depth at middle, and the quotient will be the area in inches 
of the bottom flange ; divide this area by the width of the 
bottom flange, and the quotient will be its thickness. Of the 
top flange make its width equal one third that of the bot- 
tom flange, and its thickness equal to three quarters that of 
the latter. Make the thickness of the web equal to f that 
of the bottom flange. 

Example. — What should be the dimensions of the cross- 
section of a cast-iron girder 20 feet long between bearings, 
and 24 inches high at middle, where 30,000 pounds is to be 
carried ; the factor of safety being put at 5 ? 

Here we have W = 30000 ; a := c^ ; / = 20 ; and d = 
24 ; therefore, by the rule — 

30000 X 5 X 20 



THE BOWSTRING GIRDER. 



163 



This is the area of the bottom flange. If the width of this 
flange be 12 inches, then 25.773 divided by 12 gives 2- 15, or 
2^ full, as the thickness. One third of 12 equals 4, equals 
the width of the top flange ; and | of 2-15 equals i- 61, or if— 
its thickness. The thickness of the Aveb equals f x 2 • 1 5 = 
I '34 or i^ inches. 

192.— Cast-Iron Girder: I^oad Uniformly Distributed.- 

The equation suitable to this is — 



a — 



U al 
9700 cC 



>.) 



a rule of like form with that of the last article ; therefore, 
Rule LXIV. may be used for this case, simply by substitut- 
ing 9700 for 4850, 

193.— Cast-iron Bowstring^ Girder.— An arched girder, 
such as that in Fig. 48, is technically termed a '' bowstring 
girder." The curved part is a cast-iron beam of T form in 
section, and the horizon- 
tal line is a wrought-iron 
tie-rod attached to the 
ends of the arch. This 
girder has but little to 
commend it, and is by no 
means worthy the confi- 
dence placed in it by 
builders, with many of Avhom it is quite popular. The brick 
arch usually turned over it is adequate to sustain the entire 
compressive force induced from the load (the brick wall 
built above it), and it thereby supersedes the necessity for the 
iron arch, which is a useless expense. The tie-rod is the 
only useful part of the bowstring girder, but it is usually 
made too small, and not infrequently is seriously injured by 
the needless strain to which it is subjected when it is 
'* shrunk in" to the sockets in the ends of the arch. The bow- 
string girder, therefore, should never be used. 

1 94-.— Substitute for the Bowstring Girder. — As the cast- 
iron arch of a bowstring girder serves only to resist com- 




164 



CONSTRUCTION. 



pression, its place can as well be filled by an arch of brick, 
footed on a pair of cast-iron skew-backs ; and these held 
in position by a pair of wrought-iron tie-rods, as shown in 
Fig. 49. This system of construction is preferable to the 

bowstring girder, in that the 
tie-rods are not liable to injur)' 
by '' shrinking in," and the 
cost is less. From Transverse 
StraiiiSy Art. 596, we have — 




D 



/- 



U I 



9425 d 



(89.) 



Fig. 49. 



an equation in which D is the 
diameter in inches of each of 
the two tie-rods of the brick 
arch ; U is the load in pounds 
uniformly distributed over the arch ; / is the span of the 
arch in feet ; and d, in inches, is its versed sine, or its height 
measured from the centre of the tie-rod to the centre of the 
thickness or height of the arch at middle. 

This equation may be put in words as folloAvs : 
Ride LXV. — Multiply the weight by the length ; divide 
the product by 9425 times the depth, and the square root of 
the quotient will be the diameter of each rod. 

Example. — What should be the diameter of each of the 
pair of tie-rods required to sustain a brick arch 20 feet span 
from centres, with a versed sine or height at middle of 30 
inches, to carry a brick wall 12 inches thick and 30 feet high, 
weighing 100 pounds per cubic foot? The load upon this 
arch will be for so much of the wall as will occur over the 
opening, which will be about one foot less than the span of 
the arch, or 20 — i = 19 feet. Therefore, the load will 
equal 19 x 30X i x 100 r= 57,000 pounds ; and hence, U = 
57000, / — 20, d = 30, and, by the rule — 



n 



a/ 57000 X 20 ,/ — „ 

—V — ■ = y 4-0318 = 2-008; 

9425 X 30 



or, the diameter of each rod is required to be 2 inches. 



STRAINS REPRESENTED GRAPHICALLY. 



65 



FRAMED GIRDERS. 

(95- — Graphic Representation of Strains. — In the first 
part of this section, commencing at Art. 71, the method was 
developed of ascertaining the strains in the various parts of 
a frame by the parallelogram or triangle of forces. The 
method, so far as there explained, is adequate to solve sim- 
ple cases ; but when more than three pieces of a frame con- 
verge in one point, the task by that method becomes difficult. 
This difficulty, however, disappears when recourse is had to 
the method known as that of *' Reciprocal Figures, Frames, 
and Diagrams of Forces," proposed by Professor I. Clerk 
Maxwell in 1867. This is an extension of the method by 
the triangle of forces, and may be illustrated as follows : 





Fig. 50. 

Let the lines in Fig. 50 represent, in direction and 
amount, four converging forces in equilibrium in any frame, 
as, for example, the truss of a roof; let the lines in Fig. 51 
be drawn parallel to those in Fig. 50, in the manner fol- 
lowing, namely : Let the line A B be drawn parallel with the 
line of Fig. 50 which is between the corresponding letters 
A and B, and let it be of corresponding length ; from B draw 
the line B C parallel with the line of Fig. 50 which is be- 
tween the letters B and C, and of corresponding length ; 
then from C draw C Z>, and from A draw A D, respectively 
parallel with the lines of Fig. 50 designated by the corre- 
sponding letters, and extend them till they intersect at D. 
The lengths of these two lines, the last two drawn, are de- 
termined by the point D where they intersect ; their lengths, 
therefore, need not be previously known. The lengths of 
the lines in Fig. 5 1 are respectively in proportion to the 



1 66 CONSTRUCTION. 

several strains in Fig. 50, provided these strains are in 
equilibrium. Fig. 51 is termed a closed polygon of forces. 
A system of such polygons, one for each point, in the frame 
where forces converge, so constructed that no line repre- 
senting a force shall be repeated, is termed a diagram of 
forces. This diagram of forces is a reciprocal of the frame 
from which it is drawn, its lines and angles being the same. 
The facility of tracing the forces in the diagram of forces 
depends materially upon the system of lettering here shown, 
and which was proposed by Mr. Bow, in his excellent work 
on the Ecofiouiics of Construction. In this system each 
line of the frame is designated by the two letters which it 
separates ; thus the line between A and B is called line A B ; 
that between C and D is called line CD ; and so of others ; 
and in the diagram the corresponding lines are called by the 
same letters, but here the letters designating the line are, as 
usual, at the ends of the line. Any point in a frame where 
forces converge is designated by the several letters which 
cluster around it; as, for example, in Fig. 50, the point of 
convergence there shown is designated as point A B C D. 

This invaluable method of defining graphically the 
strains in the various pieces composing a frame, such as a 
girder or roof-truss, is remarkably simple, and is of general 
application. Its utility will now be exemplified in its appli- 
cation to framed girders, and afterwards to roof-trusses. 

196.— Framed Girders. — Girders of solid timber are use- 
ful for the support of floors only where posts are admissible 
as supports, at intervals of from 8 to 15 feet. For unob- 
structed long spans it becomes requisite to construct a frame 
to serve as a girder (Arts. 163, 182). A frame of this kind 
requires two horizontal pieces, a top and a bottom chord, 
and a system of struts and suspension-pieces by which 
the top and bottom chords are held in position, and the 
strains from the load are transmitted to the bearings at the 
ends of the girders. Various methods of arranging these 
struts and ties have been proposed. One of the most simple 
and effective is shown in Fig. 52, forming a series of isos- 
celes triangles. The proportion between the length and 
height of a girder is important as an element of economy 



d^—-', (90.) 



RULES FOR FRAMED GIRDERS. 1 6/ 

both of space and cost. When circumstances do not control 
in limiting the height, it may be determined by this equation 
from Transverse Strains, Art. 624 — 

(1 75+^) /. 
2400 ' 

in which d is the depth or height between the axes of the 
top and bottom chords, and / is the length between the cen- 
tres of bearings at the supports {d and / both in feet). This 
equation in words is as follows : 

Rule LXVI. — To the length add 175 ; multiply the sum 
by the length ; divide the product by 2400, and the quotient 
will be the required height between the axes of the top and 
bottom chords. 

Exa7nple.—\Yh2it should be the depth of a girder which 
is 40 feet long between the centres of action at the supports ? 
For this the rule gives — 

^^ (i7S+40)x40 ^ 

2400 J D A ^ 

or, the proper depth for economy of material is 3 feet and 
7 inches. 

The number of bays, panels, or triangles into which the 
bottom chord may be divided is a matter of some considera- 
tion. Usually girders frorn — 

20 to 59 feet long should have 5 bays. 

59 *' 85 '' " " " 6 '' 

85 " 107 '' '' '' "7 " 

107 '' 127 " " '' '' 8 " 

127 " 146 '' " " " 9 " 

(97-— rramed Girder and Diag^ram of Forces; — Let Fig: 
52 represent a framed girder of six bays of, say, ii feet 
each, or of a total length of 66 feet. 

The lines shown are the axial lines, or the imaginary lines 
passing through the axes of the several pieces composing the 
frame. The six arrows indicate the six pressures into which 
the equally distributed load is supposed to be divided. Each 
of these is at the apex of a triangle, the base of which lies 
along the lower chord. 



i68 



CONSTRUCTION. 



The spaces between the arrows are lettered ; so, also, the 
space between the last arrow at either end and the point of 
support has a letter, and so has each triangle, and there is 
one for the space beneath the lower chord. These letters 
are to be used in describing the diagram of forces, as was 
explained in Art, 195. The diagram of forces {Fig. 53) for 
this girder-frame is drawn as follows, namely : Upon a verti- 




52. IP 

cal line A iVmark the points A, O, P, Q, R, S, and N, at equal 
distances, to represent the six equal vertical pressures indi- 
cated by the arrows in Fig. 52. The equal distances A (9, 
OP, etc., may be made of any convenient size; but it will 



HGrF 




Fig. 53. 

serve to facilitate the measurement of the forces in the dia- 
gram if they are made by a scale of equal parts, and the 
number of parts given to each division be made equal to the 
number of tons of 2000 pounds each which is contained in 
the pressure indicated by each arrow. On this vertical line 
the distance A O represents the load at the apex of the tri- 
angle B, or the points OCB {Art. 195); the distance OP 



GRAPHICAL DIAGRAMS OF FORCES. 1 69 

represents the weight at the second arrow, or at the point 
O P'E D C, and so of the rest. If the weights upon the 
points in the upper chord had been unequal, then the divi- 
sion of the vertical line A N would have had to be corre- 
spondingly unequal, each division being laid off by the scale, 
to accord with the weight represented by each. The line 
of loads, A N, being adjusted, the other lines are drawn from 
it {Art. 195), so as to make a closed polygon for the forces 
converging at each point of the frame. Fig, 52 — commenc- 
ing with the point A B Ty Fzg. $2, where there are three 
forces, namely, the force acting through the inclined strut 
A By the horizontal force in B T, and the vertical reaction 
A T at the point of support. This last is equal to half the 
entire load, or equal to the pressure indicated by the three 
arrows, A O, O P, and P Q, and is represented in Fig. 53 by 
A Q or A T. From the point Q draw a horizontal line Q B ; 
this is parallel with the force B T oi Fig. 52, in the lower 
chord. From the point A draw A B parallel with the strut 
A B oi Fig. 52. This line intersects the line B T in B and 
closes the polygon A B TA ; the point B defines the length 
of the lines A B and B T, and these lines measured by the 
scale by Avhich the line of loads was constructed give the 
required pressures in the corresponding lines, A B and B T, 
oi Fig. 52. 

Taking next the point A B C O, where four forces meet, 
of which we already have two, namely, the force in the 
strut A B and the load A O — from the point O draw the hori- 
zontal line O C ; this is parallel to the horizontal force O C 
of Fig. 52. Now from B draw BC parallel with the suspen- 
sion-piece B C oi Fig. 52. This line intersects O C\n (7, and 
the point C limits the lines O C and B C and closes the poly- 
gon A B C O A, the four sides of which are respectively in 
proportion to the four forces converging at the point A B CO 
oi Fig. 52, and when measured by the scale by which the 
line of loads was constructed give the required strains re- 
spectively in each. Taking next the point B C B T, where 
four forces converge, of which we already have two, B C 
and B T— from B extend the horizontal line T B to D \ from 
C draw CD parallel with C D oi Fig. 52, and extend it to in- 
tersect TD in Dy and thus close the polygon. T B C D T. 



I/O CONSTRUCTION. 

The lines in a part of this polygon coincide — those from 
B to T\ this is because the two strains B 7" and D T, Fig. 52, 
He in the same horizontal line. Again, taking the point 
OCD EP, where five forces meet, three of which, O P, O C, 
and CD, we already have — draw from D the line D E parallel 
with D E oi Fig. 52, and from Pthe line PE horizontally or 
parallel with PE of Fig. 52. These two lines intersect at E 
and close the polygon P O C D E P, the sides of which meas- 
ure the forces converging in the ^omt P O C D E , Fig. 52. 
Next in order is the point DEFT, Fig. 52, where four forces 
meet, two of which, TD and D E, are known. From ^ draw 
^/^ parallel with E F in Fig. 52; and from T, TF parallel 
with T F'vcv Fig. 52 ; these two lines meet in F and close the 
polygon TD E F T, the sides of which measure the required 
strains in the hues converging at the point DEFT, Fig, 52. 
Taking next the point PEFG Q, Fig. 52, where live forces 
meet, of which we already have three, QP, PE,Sind E F— 
from jpdraw a line parallel with F G oi Fig. 52, and from Q 
a line parallel with Q G oi Fig. 52. These two intersect at G 
and complete the polygon QPE EG Q, the lines of which 
measure the forces converging at PEFG Q in Fig. 52. 

In this last polygon, a peculiarity seems to indicate an 
error: the line EG has no length ; it begins and ends at the 
same point ; or, rather, the polygon is complete without it. 
This is easily understood when it is considered that the two 
lines /^G^ and G H do not contribute any strength towards 
sustaining the loads PQ and QR, and in so far as these 
weights are concerned they might be dispensed with, and 
the space occupied by the three triangles F, G, and H left 
free, and be designated by only one letter instead of three. 
Thus it appears that there are only four instead of five forces 
at the point PEFG Q, and that the four are represented by 
the lines of the polygon QPE FQ. 

The peculiarity above explained arises from considering 
loads only on the top chord : the analysis of the case is cor- 
rect as worked from the premises given ; but in practice 
there is always more or less load on the bottom chord at the 
middle, which should be considered. This will be included 
in a case proposed in the next article. One half of the dia- 



LOAD ON BOTH CHORDS. 171 

gram of forces is now complete. The other half being ex- 
actly the same, except that it is in reversed order, need not 
here be drawn. 

198. — Framed Oirder§ : L.oad on Both Chords. — Let 

Fig. 54 represent the axial lines of a girder carrying an 




Fig. 54- 

equally distributed load on each chord, represented by the 
arrows and balls shown in the figure. Let each bay measure 
10 feet, or the length ot the girder be 50 feet, and its height 







/ 


B 




/ 


D 


X\ 


V ^ / 


/-» 


\ /\ X 




y\ " 




/ 




\ / \ / 


n 


/ \ 


/\ 


/ 




\/ \/ 




\^ / 


\/ 


\ ■ 




/\ y\ 




\y 




\ 




\ / \ 




F 




\ 


\/ 




y 




H 




"^ 



Fig. 55. 

be 4|- feet. The diagram of forces (Fig. 55) for this girder 
is obtained thus : 

The plan of the girder, Fig. 54, requires to be lettered 
as shown ; having one letter within each panel and outside 
the frame, and one between every two weights or strains. 
Then, in Fig. 55, mark the vertical X\x\q KV ^V L,M,N,0, 



172 CONSTRUCTION. 

and P, dividing' it by scale into equal parts, corresponding 
with the weights on the top chord represented by the ar- 
rows. For example, if the load at each arrow equals 6J 
tons, make K L, L M, M N, etc., each equal to 6\ parts of the 
scale. Then K P \n\\\. equal the total load on the top flange. 
Make the distance P V equal to the sum of the loads on the 
bottom chord. Then iT F equals the total load on the gir- 
der. Bisect KV\xi U\ then K U or U Fequals half the total 
load ; consequently, equals the reaction of the bearing at K 
or P of Fig. 54. 

Now, to obtain the polygon of forces converging at 
K A U, Fig. 54, we have one of these forces, K U, or the re- 
action of the bearing at KA f/, equal to K U, Fig. 55. From 
U draw UA parallel with UA of Fig. 54, and from K draw 
KA parallel with the strut KA^ Fig. 54, and intersecting the 
line UA at ^, a point which marks the limit of K A and UA, 
and closes the polygon KA UK, the sides of which are in 
proportion respectively to the three strains which converge 
at the point A UK, Fig. 54. For example, since the line K U 
by scale measures the vertical reaction, K U, of the bearing 
at A UK, Fig. 54, therefore the line K A oi the diagram of 
forces by the same scale measures the strain in the strut K A, 
Fig. 54, and the line ^ ^ of the diagram by the same scale 
measures the strain in the bottom chord at A U, Fig. 54. For 
the strains converging at K A B L, Fig. 54, of which two, 
K A and K L, are already known, we draw from A the line 
A B parallel with the line A B, Fig. 54, and from L draw L B 
parallel with L B, Fig. 54, meeting A B Sit B, a point which 
limits the two lines and closes the polygon KA B L K, the 
lines of which are in proportion respectively to the strains 
converging at the point KA B L, Fig. 54, as before explained. 
Of the five strains converging at U A B C T, we already have 
three — T U, UA,3,nd AB; to obtain the other two, make 
UQ equal to PV, equal to the total load upon the lower 
flange ; divide U Q into four equal parts, QR,RS, ST, and 
T U, corresponding with the four weights on the lower 
chord, and represented by the four balls. Fig. 54. Now, 
from T, the point marking the first of these divisions, draw 
T C parallel with T C, Fig. 54, and from B draw B C paral- 



VARIOUS STRAINS IN FRAMED GIRDERS. 1 73 

lei with the strut B C, Fig, 54, meeting TC in Cj a point 
which limits the lines B C and TC and closes the polygon 
T UA BC T, the sides of which are in proportion respectively 
to the strains converging in the point T UA B C 7", Fig. 54. 
Of the five forces converging at MLB CD, we already have 
three — ML, LB, and B C; to obtain the other two, from M 
draw ML) parallel with ML>, Fig. 54, and from C draw CLJ 
parallel with CL>, Fig. 54, meeting ML) at D, a point Hmit- 
ing the lines M L> and CD and closing the polygon 
MLB CD M, the sides of which are in proportion to the 
strains converging at the point MLB CD, Fig. 54. Of the 
five forces converging at the point 5 T C D E, three — 5 T, 
T Cj and CD — are known ; to obtain the other two, from 5 
draw SE parallel with SE, Fig. 54, and from D draw D E^ 
parallel with the strut D E, Fig. 54, meeting the line S E in 
E, a point limiting the two lines S E and D E and closing the 
polygon 5 TC D ES, the sides of which are in proportion to 
the strains converging at vS TCD E, Fig. 54. One half of the 
strains in Fig. 54 are now shown in its diagram of forces, Fig. 
55 ; and since the two halves of the girder are symmetrical, 
the forces in one half corresponding to those in the other, 
hence the Hues of the diagram for one half of the forces 
may be used for the corresponding forces of the other half. 

199. — Framed Girders: Dimensions of Parts. — The 

parts of a framed girder are the two horizontal chords (top 
and bottom) and the diagonals — the struts and ties. The top 
chord is in a state of compression, while the bottom chord 
experiences a tensile strain. Those of the diagonal pieces 
which have a direction from the top to the bottom chord, 
and from the middle towards one of the bearings of the 
girder, as KA, B C, or D E, Fig. 54, are struts, and are sub- 
jected to compression. The diagonal pieces which have a 
direction from the bottom to the top chord, and from the 
middle towards one of the supports, as A B or C D, Fig. 54, 
are ties, and are subjected to extension, {Art. 83). The 
amount of strain in each piece in a framed girder having 
been ascertained in a diagram of forces, as shown in Arts. 
197 and 198, the dimensions of each piece may be obtained 



174 CONSTRUCTION. 

by rules already given. The dimensions of the pieces in a 
state of compression are to be ascertained by the rules for 
posts in Arts. 107 to 114, and those in a state of tension by 
Arts. 117 to 119 (see Arts. 226 to 229). Care is required, in 
obtaining the size of the lower chord, to allow for the joints 
which necessarily occur in long ties, for the reason that tim- 
ber is not readily obtained sufficiently long without splicing. 
Usually, in cases where the length of the girder is too great 
to obtain a bottom chord in one piece, the chord is made up 
of vertical lamina, and in as long lengths as practicable, and 
secured with bolts. A chord thus made will usually require 
about twice the material ; or, its sectional area of cross-sec- 
tion will require to be twice the size of a chord which is in 
one whole piece ; and in this chord it is usual to put the fac- 
tor of safety at from 8 to 10. 

The diagonal ties are usually made of wrought iron, and 
it is well to secure the struts, especially the end ones, with 
iron stirrups and bolts. And, to prevent the evil effects of 
shrinkage, it is well to provide iron bearings extending 
through. the depth of each chord, so shaped that the struts 
and rods may have their bearings upon it, instead of upon 
the wood. 

PARTITIONS. 

200. — PartUloii§. — Such partitions as are required for 
the divisions in ordinary houses are usually formed by tim- 
ber of small size, termed stitds or joists. These are placed 
upright at 12 or 16 inches from centres, and well nailed. 
Upon these studs lath are nailed, and these are covered 
with plastering. The strength of the plastering depends in 
a great measure upon the cHnch formed by the mortar which 
has been pressed through between the lath. That this 
clinch may be interfered with in the least possible degree, it 
is proper that the edges of the partition-joists which are 
presented to receive the lath should be as narrow as prac- 
ticable ; those which are necessarily large should be reduced 
by chamfering the corners. The derangements in floors, 
plastering, and doors which too frequently disfigure the 
interior of pretentious houses with gaping cracks in the 



FRAMED TARTITIONS. 



75 



plastering and in the door-casings are due in nearly all cases 
to defective partitions, and to the shrinkage of floor-timbers. 
A plastered partition is too heavy to be trusted upon an ordi- 
nary tier of beamsj unless so braced as to prevent its weight 
from pressing upon the beams. This precaution becomes es- 
pecially important when, in addition to its own weight, the 
partition serves as a girder to carry the weight of the floor- 
beams next above it. In order to reduce to the smallest 
practicable degree the derangements named, it is important 
that the studs in a partition should be trussed or braced so 
as to throw the weight upon firmly sustained points in the 
construction beneath, and that the timber in both partitions 
and floors should be well seasoned and carefully framed. 
To avoid the settlement due to the shrinkage of a tier of 
beams, it is important, nia partition standing over one in the 
story below or over a girder, that the studs pass between 
the beams to the plate of the lower partition, or to the 
girder ; and, to be able to do this, it is also important to ar- 
range the partitions of the several stories vertically over 
each other. All principal partitions should be of brick, 
especially such as are required to assist in sustaining the 
floors of the building. 



r~-l 

1 








^ 


; 1 


-^// 


1 


1 ' 
1 i 

' 1 1 


l! 




1( ^ 

' !' 

i 

1 


V 1 1 1 1 


^ 


J 








U 



Fig. 56. 



201. — Examples of Partitions. — Fig. 56 represents a par- 
tition having a door in the middle. Its construction is simple 
but effective. Fig. 57 shows the manner of constructing a 



176 



CONSTRUCTION. 



partition having doors near the ends. The truss is formed 
above the door-heads, and the lower parts are suspended 
from it. The posts a and b are halved, and nailed to the 
tie c d and the sill ef. The braces in a trussed partition 





iS^.-^ 



Fig. 57. 

should be placed so as to form, as near as possible, an angle 
of 40 degrees with the horizon. The braces in a partition 
should be so placed as to discharge the weight upon the 



P^. 



R^ 



\-Y/ 



^, 



r 



^ 



1^ 



y 



Fig. 58. 



points of support. All oblique pieces that fail to do this 
should be omitted. 

When the principal timbers of a partition require to be 
large for the purpose of greater strength, it is a good plan 



WEIGHT UPON PARTITIONS. 1 77 

to omit the upright filling-in pieces, and in their stead to 
place a few horizontal pieces, as in Fig. 58, in order that upon 
these and the principal timbers upright battens may be 
nailed at the proper distances for lathing. A partition thus 
constructed requires a little more space than others ; but it 
has the advantage of insuring greater stability to the plas- 
tering, and also of preventing to a good degree the conver- 
sation of one room from being overheard in the adjoining 
one. Ordinary partitions are constructed with 3x4, 3x5, 
or 4 X 6 inch joists, for the principal pieces, and with 2x4, 
2x5, or 2x6 filling-in studs, well strutted at intervals of 
about 5 feet. When a partition is required to support, in 
addition to its own weight, that of a floor or some other 
burden resting upon it, the dimensions of the timbers should 
be ascertained, by applying the principles which regulate 
the laws of pressure and those of the resistance of timber, 
as explained in the first part of this section, and in Arts. 196 
to 199 for framed girders. The following data may assist in 
calculating the amount of pressure upon partitions : 

White-pine timber weighs from 22 to 32 pounds per cubic 
foot, varying in accordance with the amount of seasoning it 
has had. Assuming it to weigh 30 pounds, the weight of 
the beams and floor-plank in every superficial foot of the 
flooring w^ill be — 

6 pounds when the beams are 3x8 inches, and placed 20 inches from centres 
7i " " " 3 X 10 " " iS 

9 " " " 3 X 12 " " 16 " " 

II " " " 3x12 " " 12 " " 

13 " " " 4x12 " " 12 " " 

13 " " " 4x14 *' " 14 

In addition to the beams and plank, there is generally the 
plastering of the ceiling of the apartments beneath, and some- 
times the deafening. Plastering may be assumed to weigh 9 
pounds per superficial foot, and deafening 1 1 pounds. 

Hemlock weighs about the same as white pine. A parti- 
tion of 3x4 joists of hemlock, set 12 inches from centres, 
therefore, will w^eigh about 2\ pounds per foot superficial 
and when plastered on both sides, 20J- pounds. 



178 



CONSTRUCTION. 



ROOFS. 

202. — Roofs. — In ancient Norman and Gothic buildings, 
the walls and buttresses Avere erected so massive and firm 
that it was customary to construct their roofs without a tie- 
beam, the walls being abundantly capable of resisting the 
lateral pressure exerted by the rafters. But in modern 
buildings, usually the walls are so slightly built as to be in- 
capable of resisting much if any oblique pressure ; hence 
the necessity of care in constructing the roof so as to avoid 
oblique and lateral strains. The roof so constructed, instead 
of tending to separate the walls, will bind and steady them. 






Fig. 59. 



Fig. 60. 



Fig. 61. 






Fig. 62. 



Fig. 63. 



Fig. 64. 






Fig. 65. 



Fig. 66. 



Fig. 67. 



203. — Comparison of Roof-Trii§sc§. — Desipfus for roof- 
trusses, illustrating various principles of roof construction, 
are herewith presented. 

The designs at Figs. 59 to 63 are distinguished from those 
at Figs. 64 to 6y by having a horizontal tie-beam. In the 
latter group, and in all designs similarly destitute of the 
horizontal tie at the foot of the rafters, the strains are much 
greater than in those having the tie, unless the truss be pro- 



VARIOUS FORMS OF ROOF-TRUSSES. 179 

tected by exterior resistance, such as may be afforded by 
competent buttresses. 

To the uninitiated it may appear preferable, in Fig. 64, 
to extend the inchned ties to the rafters, as shown by the 
dotted lines. But this would not be beneficial ; on the con- 
trary, it would be injurious. The point of the rafter where 
the tie would be attached is near the middle of its length, 
and consequently is a point the least capable of resisting- 
transverse strains. The weight of the roofing itself tends to 
bend the rafter ; and the inclined tie, were it attached to the 
rafter, would, by its tension, have a tendency to increase this 
bending. As a necessary consequence, the feet of the rafters 
would separate, and the ridge descend. 

In Fig. 65 the inclined ties are extended to the rafters ; 
but here the horizontal strut or straining beam, located at 
the points of contact between the ties and rafters, counteracts 
the bending tendency of the rafters and renders these points 
stable. In this design, therefore, and only in such designs, it 
is permissible to extend the ties through to the rafters. 
Even here it is not advisable to do so, because of the in- 
creased strain produced. (See Figs. 77 and 79.) The design 
in Fig, 64, 66, or 6j is to be preferred to that in Fig. 65. 

204-.— rorce Diag^ram : L<oad upon £acli Support.— By a 

comparison of the force diagrams hereinafter given, of each 
of the foregoing designs, we«may see that the strains in the 
trusses without horizontal tie-beams at the feet of the rafters 
are greatly in excess of those having the tie. In constructing 
these diagrams, the first step is to ascertain the reaction of, 
or load carried by, each of the supports at the ends of the 
truss. In symmetrically loaded trusses, the weight upon 
each support is always just one half of the whole load. 

205 -—Force Diagram for Trus§ in Fig. 59. — To obtain 
the force diagram appropriate to the design in Fig. 59, first 
letter the figure as directed in Art. 195, and as in Fig. 68. 
Then draw a vertical X\wq,E F {Fig. 69), equal to the weight W 
at the apex of roof ; or (which is the same thing in effect) 
equal to the sum of the two loads of the roof, one extending 
on each side of f'F half-way to the foot of the rafter. Di- 



i8o 



CONSTRUCTION. 



vide E F into two equal parts at G. Make G C and G D eaOi 
equal to one half of the weight N. Now, since E G l"^ equal 
to one half of the upper load, and G D to one half of the low- 
er load, therefore their sum, EG+ G D = ED, is equal to 
one half of the total load, or to the reaction of each support, 
E or F. From D draw DA parallel with DA of Fig. 68, and 
from E draw E A parallel with E A oi Fig. 6%. The three 
lines of the triangle A E D represent the strains, respectively, 
in the three lines converging at the point A D E oi Fig. 68. 
Draw the other lines of the diagram parallel with the lines of 





Fig. 69. 



Fig. 68, and as directed in Arfs. 195 and 197. The various 
lines of Fig. 69 will represent the forces in the corresponding 
lines of Fig. 68 ; bearing in mind {Art. 195.) that while a line 
m the force diagram is designated in the usual manner by the 
letters at the two ends of it, a line of the frame diagram is 
designated by the two letters between which it passes. Thus, 
the horizontal lines A D, the vertical lines A B, and the in- 
clined lines A E have these letters at their ends in Fig. 69, 
while they pass between these letters in Fig. 68. 

206.— Force I$la^raiii forTru§§ in Fig. 60. — For this truss 
we have, in Fig. 70, a like design, repeated and lettered as 
required. We here have one load on the tie-beam, and three 
loads above the truss: one on each -rafter and one at the 
ridge. In the force diagram, Fig. 71, make G H, H J, and 
J K, by any convenient scale, equal respectively to the 
weights GH, HJ, and J K oi Fig. 70. Divide GK'wXo two 
equal parts at L. Make LE and Z/^each equal to one half 
the weight E F {Fig. 70). Then G F is equal to one half the 



FORCE DIAGRAMS OF TRUSSES. 



l8l 



total load, or to the load upon the support G (Art. 205V 
Complete the diagram by drawing its several lines parallel 
with the lines oi Fig. 70, as indicated by the letters (see Art. 
205), commencing with G F, the load on the support G {Fig. 
70). Draw from F and G the two lines FA and G A paral- 
lel with these lines in Fig. yo. Their point of intersection 
defines the point A. From this the several points B, C, and 
D are developed, and the figure completed. Then the lines 
in Fig. 71 will represent the forces in the corresponding lines 
of Fig. 70, as indicated by the lettering. (See Art. 195.) 




Fig. 70. 



207. — Force Diagrram for Truss in Fig 61. — For this truss 
we have, in Fig. 72, a similar design, properly prepared by 
weights and lettering ; and in Fig. 73 the force diagram ap- 
propriate to it. 

In the construction of this diagram, proceed as directed 
in the previous example, by first constructing N Sy the ver- 
tical line of weights ; in which line NO.OP, PQ,QR, and R S 
are made respectively equal to the several weights above 
the truss in Fig. 72. Then divide NS into two equal parts at 7\ 
Make T^/iTand TL each equal to the half of the weight K L. 
Make J K and L M equal to the weights J K and L M oi Fig. 
72. Now, since 3fN is equal to one half of the weights above 
the truss plus one half of the weights below the truss, or half of 
the whole weight, it is therefore the weight upon the support 
N {Fig. 72), and represents the reaction of that support. A 
horizontal line drawn from M will meet the inclined line 
drawn from iV, parallel with the rafter A N {Fig. 72), in the 



l82 



CONSTRUCTION. 



points, and the three sides of the triangle A MN, Fig. 73, will 
give the strains in the three corresponding lines meeting at 
the point A MN, Fig. 72. The sides of the triangle HjfS, Fig. 




Fig. 72. 

73, give likewise the strains in the three corresponding lines 
meeting at the point H J S, Fig.ji. Continuing the con- 
struction, draw all the other lines of the force diagram parallel 




Fig. 73- 

with the corresponding lines of Fig. 72, and as directed in 
Art. 195. The completed diagram will measure the strains 
in all the lines of Fig. 72. 



FORCE DIAGRAMS CONTINUED. 



83 



208. — Force Diag^raiii for Tru§s in Fig. 63. — The roof 
truss indicated at Fig, 63 is repeated in Fig. 74, with the ad 




Fig. 74. 




Fig. 75. 

dition of the lettering required for the construction of the 
force diagram, Fig. 75. 



1 84 CONSTRUCTION. 

In this case there are seven weights, or loads, above the 
truss, and three below. Divide the vertical line O V 3.t W 
into two equal parts, and place the lower loads in two equal 
parts on each side of W. Owing to the middle one of these 
loads not being on the tie-beam with the other two, but on 
the upper tie-beam, the line G H, its representative in the 
force diagram, has to be removed to the vertical BJ, and 
the letter AT is duplicated. The line NO equals half the 
whole weight of the truss, or 3^ of the upper loads, plus one 
of the lower loads, plus half of the load at the upper tie- 
beam. It is, therefore, the true reaction of the support NO^ 
and A N \s the horizontal strain in the beam there. It will 
be observed also that while H M and G M {Fig. 75), which 
are equal lines, show the strain in the lower tie-beam at the 
middle of the truss, the lines C H and FG, also equal but 
considerably shorter lines, show the strains in the upper 
tie-beam. Ordinarily, in a truss of this design, the strain in 
the upper beam would be equal to that in the lower one, 
Avhich becomes true when the rafters and braces above the 
upper beam are omitted. In the present case, the thrusts of 
the upper rafters produce tension in the upper beam equal 
to 6^^^ or F3I of Fig. 75, and thus, by counteracting the 
compression in the beam, reduce it to CH or FG of the 
force diagram, as shown. 



209, — Force Diag^ram for Tru§s in Fig. 64. — The force 
diagram for the roof-truss at Fig. 64 is given in Fig. yj, 
while Fig. 78 is the truss reproduced, with the lettering 
requisite for the construction oi Fig. yy. 

The vertical FF {Fig. yy) represents the load at the 
ridge. Divide this equally at W, and place half the lower 
weight each side of W, so that CD equals the lower 
weight. Then FD is equal to half the whole load, and 
equal to the reaction of the support E {Fig. y6). The lines 
in the triangle A D E give the strains in the corresponding 
lines converging at the point A D E oi Fig. yo. The other 
lines, according to the lettering, give the strains in the cor- 
responding lines of the truss. (See Art. 195.) 



FORCE DIAGRAMS CONTINUED. 



[85 



2(0.— Force Diagram for Truss in Fig. 65. — This truss is 
reproduced in Fig, 78, with the letters proper for use in the 
force diagram, Fig. 79. 




Fig. 79. 

Here the vertical G K, containing the three upper loads 
GH, H y, and J K, is divided equally at IV, and the lower 



1 86 



CONSTRUCTION. 



load E Fis placed half on each side of W, and extends from £ 
to F. Then FG represents one half of the whole load of the 
truss, and therefore the reaction of the support G {Fig. 78). 
Drawing the several lines of Fig. 79 parallel with the corre- 
sponding lines of Fig. 78, the force diagram is complete, and 
the strains in the several lines of 78 are measured by the cor- 
responding lines of 79. (See Art. 195.) 

A comparison of the force diagram of the truss in Fig. j6 
with that of the truss in Fig. 'jZ shows much greater strains 
in the latter, and we thus see that Fig. ^6 or 64 is the more 
economical form. 




Fig. 80. 



211. — Force Diagram for Truss in Fig. 66. — This truss is 
reproduced and prepared by proper lettering in Fig. 80, and 
its force diagram is given in Fig. 81. 

Here the vertical JM contains the three upper loads 
yK, KL, and LM. Divide jf M into two equal parts at 
Gy and make FG and G H respectively equal to the two 
loads FG and G H oi Fig. 80. Then H J represents one 
half of the whole weight of the truss, and therefore the reac- 
tion of the support J. From H and J draw lines par- 
allel with A H and A J oi Fig. 80, and the sides of the tri- 
angle A H y will give the strains in the three lines concen- 
trating in the point A H y {Fig. 80). The other lines of Fig. 



EFFECT OF ELEVATING THE TIE-BEAM, 



187 



81 are all drawn parallel with their corresponding lines in 
Fig. 80, as indicated by the lettering. (See Art. 195.) 




Fig. 81. 



212. — Roof-Trus§: Effect of Elevating the Tie-Beam. — 

From Arts. 670, 671, Transverse Strains^ it appears that the 




Fig. 82 



effect of substituting inclined ties for the horizontal tie at 
feet of rafters is — 



1 88 CONSTRUCTION. 

F=P| (91.) 

in which P represents half the weight of the whole truss and 
the load upon it ; a + b — height of the truss at middle above 
a horizontal line drawn at the feet of the rafters ; a equals 
the height from this line to the point where the two inclined 
ties meet ; b, the height thence to the top of the truss ; and 
V, the additional vertical strain at the middle of the truss 
due to elevating the tie from a horizontal line. 

Examples are given to show that when the elevation of 
the tie equals J of the whole height, the vertical strain there- 
by induced is equal to a weight which equals J of half the 
whole load ; and that when the elevation equals half the 
whole height, the vertical strain is equal to half the whole 
load. This is the strain in the vertical rod at middle. The 
strains in the rafters and inclined ties are proportionately 
increased. 

213. — Planning a Roof. — In designing a roof for a build- 
ing, the first point requiring attention is the location of the 
trusses. These should be so placed as to secure solid bear- 
ings upon the walls ; care being taken not to place either of 
the trusses over an opening, such as those for windows or 
doors, in the wall below. Ordinarily, trusses are placed so 
as to be centrally over the piers between the windows ; the 
number of windows consequently ruling in determining the 
number of trusses and their distances from centres. This 
distance should be from ten to twenty feet ; fifteen feet apart 
being a suitable medium distance. The farther apart the 
trusses are placed, the more they will have to carry ; not 
only in having a larger surface to support, but also in that 
the roof-timbers will be heavier ; for the size and weight of 
the roof-beams will increase with the span over which they 
have to reach. 

In the roof-covering itself, the roof-planking may be laid 
upon jack-rafters, carried by purlins supported by the 
trusses ; or upon roof-beams laid directly upon the back of 
the principal rafters in the trusses. In either case, proper 



LOAD UPON ROOFS. 1 89 

struts should be provided, and set at proper intervals to re- 
sist the bending of the rafter. In case purlins are used, one of 
these struts should be placed at the location of each purlin. 

The number of these points of support rules largely in 
determining the design for the truss, thus : 

For a short span, where the rafter Avill not require sup- 
port at an intermediate point, Fig. 59 or 64 will be proper. 

For a span in which the rafter requires supporting at one 
intermediate point, take Fig. 60, 65, or 66. 

For a span with two intermediate points of support for 
the rafter, take Fig. 61 or 6^. 

For a span with three intermediate points, take Fig. 63. 

Generally, it is found convenient to locate these points of 
support at nine to twelve feet apart. They should be suffi- 
ciently close to make it certain that the rafter will not be sub- 
ject to the possibility of bending. 

214. — Load upon Moof-TrM§§. — In constructing the force 
diagram for any truss, it is requisite to determine the points 
of the truss which are to serve as points of support (see 
Figs. 70, 72, etc.), and to ascertain the amount of strain, or 
loading, which will occur at every such point. 

The points of support along the rafters will be required 
to sustain the roofing timbers, the planking, the slating, the 
snow, and the force of the wind. The points along the tie- 
beam will have to sustain the weight of the ceiling and the 
flooring of a loft within the roof, if there be one, together 
with the loading upon this floor. The weight of the truss 
itself must be added to the weight of roof and ceiling. 

215.— Lioad on Roof per Superficial Foot. — In any im- 
portant work, each of the items in Art. 214 should be care- 
fully estimated, in making up the load to be carried. For 
ordinary roofs, the weights may be taken per foot superficial, 
as follows : 

Slate, about j-o pounds. 

Roof-plank, '' 2-7 

Roof-beams or jack-rafters, '' 2-3 " 

In all, 12 pounds. 



IQO CONSTRUCTION. 

This is for the superficial foot of the inclined roof. For the 
foot horizontal, the augmentation of load due to the angle of 
the roof will be in proportion to its steepness. In ordinary 
cases, the twelve pounds of the inclined surface will not be 
far from fifteen pounds upon the horizontal foot. 
For the roof-load we may take as follows : 



Roofing, 


about 


15 


pounds. 


Roof-truss, 


(( 


5 


(( 


Snow, 


a 


20 


(( 


Wind, 


(< 


10 


ti 


Total on 


roof, 


50 


pounds 



per square foot horizontal. 

This estimate is for a roof of moderate inclination, say 
one in which the height does not exceed ^^ of the span. 
Upon a steeper roof the snow would not gather so heavily, 
but the wind, on the contrary, would exert a greater force. 
Again, the wind acting on one side of a roof may drift the 
snow from that side, and perhaps add it to that already 
lodged upon the opposite side. These two, the wind and 
the snow, are compensating forces. The action of the snow 
is vertical : that of the wind is horizontal, or nearly so. The 
power of the wind in this latitude is not more than thirty 
pounds upon a superficial foot of a vertical surface ; except, 
perhaps, on elevated places, as mountain-tops for example, 
where it should be taken as high as fifty pounds per foot of 
vertical surface. 

216. — Load upon Tie-Beam. — The load upon the tie- 
beam must of course be estimated according to the require- 
ments of each case. If the timber is to be exposed to view, 
the load to be carried will be that only of the tie-beam and 
the timber struts resting upon it. If there is to be a ceiling 
attached to the tie-beam, the weight to be added will be in 
accordance with the material composing the ceiling. If of 
wood, it need not weigh more than two or three pounds per 
foot. If of lath and plaster, it will weigh about nine pounds ; 
and if of iron, from ten to fifteen pounds, according to the 



WEIGHT UPON ROOFS, IN DETAIL. 



191 



thickness of the metal. Again, if there is to be a loft in the 
roof, the requisite flooring may be taken at five pounds, and 
the load upon the floor at from twenty-five to seventy 
pounds, according to the purpose for which it is to be used. 

217- — Roof-'Weiglii§ In Dciail. — The load to be sustained 
by a roof-truss has been referred to in the previous three 
articles in general terms. It will now be treated more in 
detail. But first a few words regarding the slope of the 
roof. In a severe climate, roofs ought to be constructed 
steeper than in a milder one, in order that snow may have a 
tendency to slide off before it becomes of sufficient weight 
to endanger the safety of the roof In selecting the material 
with which the roof is to be covered, regard should be had 
to the requirements of the inclination : slate and shingles 
cannot be used safely on roofs of small rise. The smallest 
inclination of the various kinds of covering is here given, 
together with the weight per superficial foot of each. 



Material. 



Tin 

Copper 

Lead , 

Zinc , 

Short pine shingles. . . 
Long cypress shingles 
Slate 



Least Inclination. 


Weight upon a 
square foot. 


Rise I inch to a foot. 

" 2 inches " " 
<< ^ (( <( i< 

.. 5 " " " 
" 6 " " " 
.. 6 " " " 


1 to li lbs. 

1 to I^ " 

4 to 7 " 
li to 2 " 
i| to 2 " 

2 to 3 " 

5 to 9 '• 



The weight of the covering as here estimated includes 
the weight of whatever is used to fix it in place, such as 
nails, etc. The weight of that which the covering is laid 
upon, such as plank, boards, or lath, is not included. The 
weight of plank is about 3 pounds per foot superficial ; of 
boards, 2 pounds ; and lath, about half a pound. 

Generally, for a slate roof, the weight of the covering, in- 
cluding plank and jack-rafters, amounts to about 12 pounds, 
as stated in Art. 215 ; but in every case, the weight of each 
article of the covering should be estimated, and the full load 
ascertained by summing up these weights. 



192 CONSTRUCTION. 

218. — Load per Foot Horizontal. — The weight ot the 
covering as referred to in the last article is the weight per 
foot on the inclined surface ; but it is desirable to know how 
much per foot, measured horizontally, this is equal to. The 
horizontal measure of one foot of the inclined surface is 
equal to the cosine of the angle of inclination. Then, to ob- 
tain the inclined measure corresponding to one foot horizon- 
tal, we have — 

cos. '. I : : p : C = ~^— ; 

COS. 

where / represents the pressure on a foot of the inclined 
surface, and C the weight of so much of the inclined cover- 
ing as corresponds to one foot horizontal. The cosine of an 
angle is equal to the base of the right-angled triangle divided 
by the hypothenuse (see Trigonometrical Terms, Art. 474), 
which in this case is half the span divided by the length ot 

the rafter, or — % , where s is the span, and / the length of the 

rafter. Hence, the load per foot horizontal equals — 

COS. s s ' (92-) 

27 

or, twice the pressure per foot of inclined surface multiphed 
by the length of the rafter and divided b}^ the span, both in 
feet, will give the weight per foot measured horizontally. 

219- — WeigBit of Truss. — The weight of the framed truss 
will be in proportion to the load and to the span. This, for 
the weight upon a foot horizontal, will about equal — 

T — 0-077 Cs', 

which equals the weight in pounds per foot horizontal to be 
allowed for a wooden truss with iron suspension-rods and a 
horizontal tie-beam, near enough for the requirements of our 
present purpose ; where s equals the length or span of the 



EFFECT OF WIND ON ROOFS. I93 

truss, and C the weight per foot horizontal of the roof cover- 
ing, as in equation (92.). Substituting for C its value, as in 
(92.), we have — 

T= 0-0077 s^i^ ; 
s 

or — 

r= 0.0154//; (93.) 

which equals the weight in pounds per foot horizontal to 
be allowed for the truss. 

220.^ — Weight of Snow on Roof^. — The weight of snow 
will be in proportion to the depth it acquires, which will be 
in proportion to the rigor of the climate of the place where 
the building is to be erected. Upon roofs of ordinary incli- 
nation, snow, if deposited in the absence of wind, will not 
slide off ; at least until after it has acquired some depth, and 
then the tendency to slide will be in proportion to the angle 
of inclination. The weight of snow may be taken, therefore, 
at its weight per cubic foot (8 pounds) multiplied by the 
depth it is usual for it to acquire. This, in the latitude of 
New York, may be taken at about 2^ feet. Its weight 
would, therefore, be 20 pounds per foot superficial, meas- 
ured horizontally. 

221. — Effect of liVind on Roof^. — The direction of wind 
is horizontal, or nearly so, when unobstructed. Precipitous 
mountains or tall buildings deflect the wind considerably 
from its usual horizontal direction. Its power usually does 
not exceed 30 pounds per superficial foot except on ele- 
vated places, where it sometimes reaches 50 pounds or more. 
This is the pressure upon a vertical surface ; roofs, however, 
generally present to the wind an inclined surface. The ef- 
fect of a horizontal force on an inclined surface is in pro- 
yjortion to the sine of the angle of inclination ; the direction 
of this effect being at right angles to the inclined surface. 
The force thus acting may be resolved into forces acting in 
two directions — the one horizontal, the other vertical ; the 
former tending, in the case of a roof, to thrust aside the walls 



94 



CONSTRUCTION. 




on which the roof rests, and the latter acting directly on the 
materials of which the roof is constructed — this latter force 
being in proportion to the sine of the angle of inclination 
multiplied by the cosine. This will be made clear by the 

following explanation. Re- 
ferring to Fig. 83, let D KE 
be the angle of inclination 
of the roof, D E being equal 
to one foot. Bisect D K at 
A ; draw A L parallel with 
Fig. 83. EK\ make A L equal to the 

horizontal pressure of the wind upon one foot superficial of 
a vertical plane. Draw A C perpendicular to D K, and LF 
parallel with A C from i^draw EC parallel with EK\ draw 
A B parallel with D E, The sides of the triangle LA F rep- 
resent the three several forces in equilibrium : L A \^ the 
force of the wind ; L E is the pressure upon the roof ; and 
AE is the force with which the wind moves on up the roof 
towards D. Now, to find the relation of the force of the 
wind to the strain produced by it in the direction A C, we 
have — 



rad. 



rad. : sin. : : EC : A C; 
E C = L A ; therefore — 
: sin. : : L A : A C = LA sin. ; 
A C = E sin. ; 



or, the strain perpendicular to the surface of the roof equals 
the force of the wind multiplied by the sine of the angle of 
inclination. When A C represents this strain, then, of the 
two forces referred to above, B C represents the horizontal 
force, and A B the vertical force. To obtain this last force, 
we have — 

rad. : cos. : : A C : A B. 
Putting for A C its value as above, Ave have — 



rad. : cos. : : E s\xi. : A B = E sin. cos.; 
V = E sin. COS. ; 



FORCE OF THE WIND. I95 

or, the vertical effect is equal to the product of the force of 
the wind upon a superficial foot into the sine and the cosine 
of the angle of inclination. This result is that which is due 
to the pressure of the wind upon so much of the inclined 
surface as is covered by one square foot of a vertical sur- 
face. The wind, acting horizontally through one foot super- 
ficial of vertical section, acts on an area of inclined surface 
equal to the reciprocal of the sine of inclination, and the 
horizontal measurement of this inclined surface is equal to 
the cosine of the angle of inchnation divided by the sine. 
This may be illustrated from Fig. 83, thus — 

sin. : rad. : : D E : D K, 

D E equals i foot ; therefore — 

sin. : rad. : \ \ \ D K — —,— ; 

sm. 

or, the surface acted upon by one square foot of sectional 
area equals the reciprocal of the sine of the angle of incli- 
nation. Again, the horizontal measure of this inclined sur- 
face may be obtained thus — 

sin. : cos. \ : D E \ KE — —^ ; 

sm. 

or, KE, the horizontal measurement, equals the cosine of the 
angle of inclination divided by the sine. 

In the figure, make K G equal to one foot ; then we 
have — 

KE : KG : : V: W; 

m which V, as above, represents the vertical pressure due to 
the wind acting upon the surface KB, and IV the vertical 
pressure due to the wind acting upon the surface KH, or 
so much as covers KG, one foot horizontal. 

Now we have, as above, K E equal to ^-, K G = i, and 

sm. 



196 CONSTRUCTION. 

V = i^sin. COS. Substituting these values, we have, instead 
of the above proportion — 



sin. 
from which — 



cos. ^ . ,., 

: I : : F sin. cos. : W; 



i^sin. COS. r^ . , r \ 



Sin. 



or, the vertical effect of the wind upon so much of the roof 
as covers each square foot horizontal, is equal to the pro- 
duct of the force of the wind per square foot into the square 
of the sine of the angle of inclination. 

Example.— Wh.Qn the force of the wind upon a square 
foot of vertical surface is 30 pounds, what will be the verti- 
cal effect per square foot horizontal upon a roof the inclina- 
tion of which is 26° 33', or 6 inches to the foot? 

Here we have F — 30, and the sine of 26° 33' is 0-44698 ; 
therefore — 

W= 30x0-44698' =: 5-9937. 

This is conveniently solved by logarithms; thus— 

log. 30 = ^.4771213 
0-44698 = 9-6502868 
0-44698 = 9-6502868 

5.9937 = 0.7776949 

or, the vertical effect is (5 -9937, or) 6 pounds. 

The form of equation (94.) may be changed ; for, in a right- 
angled triangle, the sine of the angle at the base is equal to 
the perpendicular divided by the hypothenuse; which, in 
the case of a roof, is the height divided by the length of the 
rafter; or^ 

height h 
Sine = -^j-p = -. 



LOAD UPON ROOFS. I97 

Therefore, equation (94.) may be changed to — 

W=F^^', (95.) 

or, the vertical effect upon each square foot of a roof is equal 
to the product of the force of the wind per foot into the 
square of the height of the roof at the ridge, divided by the 
square of the length of the rafter (the height and length both 
in feet.) 

Example. — When the force of the wind is 30 pounds, the 
height of the roof 10 feet, and the length of the rafter 22-36 
feet, what will be the vertical effect of the wind? Here we 
have F = 30, /i = 10, and / = 22-36 ; and— 

W = 7,ox ^-^ z= 6. 

'^ 22-36' 

222. — Total Load per Foot Horizontal. — The various 
items comprising the total load upon a roof are the cover- 
ing, the truss, the wand, snow, the plastering or other kind 
of ceiling, and the load which may be deposited upon a floor 
formed in the roof ; or, the total load — 

M= C+ r+ W^-S+P+L. 

The value per foot horizontal for these has been found as 

follows: C=^ ', r= 0-0154//; W=F^. For 5 the 

value must be taken according to circumstances, as in Art. 
220. So, also, the value of P and L are to be assigned as 
required for each particular case, as in Art. 216. The total 
load, therefore, with these substitutions, will be — 

M = 1^+ 0-0154// + /^—+ 5 + P+Z; 

which reduces to — 

J/= // (-^^0'0iS4) + ^^.-\-S + P+L; (96.) 



198 CONSTRUCTION. 

in which / is the length of the rafter ; / is the weight of the 
covering per foot superficial, including the roof boards or 
slats, the jack-rafters, etc. ; s is the span of the roof ; /i is the 
vertical height above a horizontal line passing through the 
feet of the rhfters ; F is the force of the wind per square foot 
against a vertical surface ; 5 is the weight of snow per 
square foot horizontal ; P is the weight per superficial foot 
of the ceiling at the tie-beam ; and Z, the load per superficial 
foot in the roof, including weight of flooring and floor- 
timbers. The dimensions, ^, /, and //, are each in feet ; the 
weight of /, F, S, P, and L are each in pounds. The value 
of/ is for a square foot of the inclined surface. 

223. — strains in Roof-Timbers Computed.— The graphic 
method of obtaining the strains, as shown in Arts. 205 to 211, 
is, for its conciseness and simplicity, to be preferred to any 
other method ; yet, on some accounts, the method of obtain- 
ing the strains by the parallelogram of forces and by arith- 
metical computations will be found useful, and will now be 
referred to. 

By the parallelogram of forces, the weight of the roof is 
in proportion to the oblique thrust or pressure in the axis of 
the rafter as twice the height of the roof is to the length of 
the rafter ; or — 

or, transposing — 

2h:l'.:R'. Y=^', (97.) 

2/1 

where Y equals the pressure in the axis of the rafter, and R 
the weight of one truss and its load. Again, the weight of 
the roof is in proportion to the horizontal thrust in the tie- 
beam as twice the height of the roof is to half the span ; or — 



R'.H: \2h'. 



s . 

— > 



or, transposmg — 



2 /i : - : : R : H = — ^,- 
2 4/1 



THE STRAINS SHOWN GEOMETRICALLY. I99 

where // equals the horizontal thrust in the tie-beam. To 
obtain R, the weight of the roof, multiply M, the load per 
foot, as in equation (96.), by s, the span, and by c, the dis- 
tance from centres at which the trusses are placed ; or — 

R = M c s. 

With this value of R substituted for it, we have — 

Mcsl , 

and — 



(99.) 



.=^,f, (.00, 

in which Y equals the strain in the axis of the rafter, and H 
the strain in the tie-beam. These are the greatest strains 
in the rafter and tie-beam. At certain parts of these pieces 
the strains are less, as will be shown in the next article. 

224. — Slraiii§ in Roof-Timbers Shown Geometrically. — 

The pressure in each timber may be obtained as shown in 
Fig. 84, where A B represents the axis of the tie-beam, A C 
the axis of the rafter, D E and FB the axes of the braces, 
and DG^ F E, and C B the axes of the suspension-rods. In 
this design for a truss, the distance A B is divided into three 
equal parts, and the rods located at the two points of division, 
G and E. By this arrangement the rafter A Cis supported at 
equidistant points, B and F. The point D supports the rafter 
for a distance extending half-way to A and half-way to F, and 
the point F sustains half-way to D and half-way to C. Also, 
the point C sustains half-way to F, and, on the other rafter, 
half-way to the corresponding point to F. And because these 
points of support are located at equal distances apart, there- 
fore the load on each is the same in amount. On D G make 
Da equal by any decimally divided scale to the number of 
hundreds of pounds in the load on Z>, and draw the parallel- 
ogram abDc. Then, by the same scale, Db represents 
{Art. 71) the pressure in the axis of the rafter by the load at 



200 



CONSTRUCTION, 




Fig. 84. 



STRAINS IN A TRUSS. 201 

D\ also, Z^^ the pressure in the brace /)^. Draw <:<^ hori- 
zontal ; then D d'ls the vertical pressure exerted by the brace 
D E Sit E. The point /^sustains, besides the common load 
represented by Da, also the vertical pressure exerted by the 
brace DE\ therefore, make Fe equal to the sum oi D a and 
Dd, and draw the parallelogram F gef. Then Fg, meas- 
ured by the scale, is the pressure in the axis of the rafter 
caused by the load at F, and Ff is the load in the axis of the 
brace FB. Draw /// horizontal ; then Fk is the vertical 
pressure exerted by the brace F B2X B. The point C, besides 
the common load represented by D a, sustains the vertical 
pressure Fh caused by the brace FB, and a like amount 
from the corresponding brace on the opposite side. There- 
fore, make Cj equal to the sum oi D a and twice F h, and 
draw j k parallel to the opposite rafter. Then Ck is the 
pressure in the axis of the rafter at C. This is not the only 
pressure in the rafter, although it is the total pressure at its 
head C. At the point Z', besides the pressure (^/', there is 
F g. i\t the point Z>, besides these two pressures, there is 
the pressure D b. At the foot, at A, there is still an addi- 
tional pressure ; for while the point D sustains the load half- 
way to F and half-way to A, the point A sustains the load 
half-way to D. This load is, in this case, just half the load 
at D. Therefore draw A vi vertical, and equal, by the scale, 
to half of Da, Extend CA to/; draw ml horizontal. 
Then ^ / is the pressure in the rafter at A caused by the 
weight of the roof from A half-way to D. Now the total of 
the pressures in the rafter is equal to the sum oi A 1+ D b + 
/^^ added to C k. Therefore make kn equal to the sum of 
A l+Db + Fg, and draw no parallel with the opposite raf- 
ter, and nj horizontal. Then Co, measured by the same 
scale, will be found equal to the total weight of the roof on 
both sides of C B. Since Da represents s, the portion of the 
weight borne by the point D, therefore Co, representing the 
whole weight of the roof, should equal six times Da, as it 
does, because D supports just one sixth of the whole load. 
Since C n is the total oblique thrust in the axis of the rafter 
at its foot, therefore nj is the horizontal thrust in the tie- 
beam at A, 



202 CONSTRUCTION. 

225.— Application of tlie Geometrical System of Strains.— 

The strains in a roof-truss can be ascertained geometrically, 
as shown in Art. 224. To make a practical application of 
the results, in any particular case, it is requisite first to as- 
certain the load at the head of each brace, as represented by 
the line D a, Fig. 84. The load corresponding to any part 
of the roof is equal to the product of the superficial area of 
that particular part (measured horizontally) multiplied by 
the weight per square foot of the roof. Or, when M equals 
the weight per square foot, c the distance from centres at 
which the trusses are placed, and 71 the horizontal distance 
between the heads of the braces, then the total load at the 
head of a brace is represented by — 

N=Mcn. (loi.) 

The value of M is given in general terms in equation (96.). 
To show its actual value, let it be required to find the weight 
per square foot upon a root 52 feet span and 13 feet high at 
middle ; or {Fig. 84), where A B equals half the space, or 26 
feet, and CB 13 feet, then/i C, the length of the rafter, will 
be 26-069, nearly. And where the weight of covering per 
square foot, on the inclination, is 12 pounds, the force of the 
wind against a vertical plane is 30 pounds ; the weight of 
snow per foot horizontal is 20 pounds ; the Aveight of the 
plastering forming the ceiling at the tie-beam is 9 pounds ; 
and the load in the roof is nothing ; — with these quantities 
substituted, equation (96.) becomes — 

M — 2q-o6qx 12 { --+ 0-0154) + 30 — ^-^ — + 20 -f 9 +0 ; 
\52 / 29-069' 

M = (29-069 X 12 X 0-05386) + (30 X 0-2) + 20 + 9; 

M = 18-788 + 6 + 29 = 53-788; 

or, say, 53-8 pounds. Then if c, the distance from centres 
between trusses, is 10 feet, and Ji, the distance between 
braces, is one third of A B, Fig. 84, or y = 8f , the total load 
at the head of a brace will be, as per equation (loi.) — 

i\^= 53-8 X 10 X 8| = 4663; 



TABLE OF STRAINS. 203 

or, say, 4650 pounds. Now, by any decimally divided scale, 
make D a, Fig. 84, equal to 46^^ parts of the scale ; this being 
the number of hundreds of pounds contained in the weight 
at D, as above. Then, by the same scale, the several lines 
in the figure drawn as before shown will be found to repre- 
sent respectively the weights here set opposite to them, as 
follows : 

Dd^^da=^he=^ 23^, and represents 2325 pounds; 

Da=zdc = hf=Fh^46^ '' 4650 

Dc = Db = Al = Fg=^2 " 5200 " 

Fe = Da +'£> d = 6gi '' 6975 

F/ = 65! '^ 6575 

0" = 3^«=i39j- " 13950 

CK= 3 Z>^ = 156 '' 15600 

C 11 — C k^ Fg^ D b-\- A l=i\2 '' 31200 " 
Cn^ Ck^lDb^ 6 Db=2Ck 

= 312 *' 31200 " 

Nj = C — 6 Da = 6x 46J = 279 '' 27900 '' 

It should be observed here that the equality of the lines 7ij 
and Co is a coincidence dependent upon the relation which 
in this particular case the line CB happens to bear to the 
line A B ; A B being equal to twice C B. And so of some 
other lines in the figure. If the inclination of the roof were 
made greater or less, the equality of the lines referred to 
would disappear. It should also be observed that the strains 
above found are not quite exact ; they are, however, correct 
to within a fraction of a hundred pounds, which is a suffi- 
ciently near approximation for the purpose intended. From 
the results obtained above, we ascertain that the strain in 
the rafter, from F to C, is represented by C K, and is equal 
to 15,600 pounds ; while the strain at the foot of the rafter, 
from A to />, is represented by C71, and equals 3 1,200 pounds, 
or double that which is at the head of the rafter. We ascer- 
tain, also, that the maximum strain in the tie-beam, repre- 
sented by nj\ is 27,900 pounds; that that in the brace £>E, 
represented by F> c, is 5200 pounds; and that that in the 
brace F B, represented by F/,is 6575 pounds. The strain 



204 CONSTRUCTION. 

in the vertical rod D G is theoretically nothing. There is, 
however, a small strain in it, for it has to carry a part of the 
tie-beam and so much of the ceiling as depends for support 
upon that part. But the manner of locating the weights, 
adopted in this article, does not recognize any load located 
at the point G. This is an objection to this system, but it 
is not material. 

For a recognition of weights at the tie-beam, see Arts. 

205 to 211. The load at G may be found by obtaining the 
product of the surface carried into the weight per foot of 
the ceiling; or, say, 10 en = 10 x lox 8f = 867 pounds. 
The load to be carried by the rod FE is shown at D d=^ ke^ 
which above is found to be 2325 pounds. To this is to be 
added 867 pounds for the ceiling at E, as before found for the 
ceiling at G; or, together, 3192 pounds. The central rod 
CB has to carry the two loads brought to B by the two 
braces footed there ; and also the weight of the ceiling sup- 
ported by B. The vertical strain from the brace F B is rep- 
resented at F/i, and equals 4650 pounds ; therefore, the 
total load on CB is 4650 + 4650 + 867 = 10,167 pounds. 

226. — Roof-Timbers: the Tie-Beam. — The roof-timbers 
comprised in the truss shown in Fi^: 84 are the rafters, 
tie-beam, two braces, and three rods. Of these, taking first 
the tie-beam, we have a piece subject to tension and some- 
times to cross-strain (see Art. 682, Transverse Strains), In 
this case the tensile strain only need be considered. For 
this a rule is given in Art. 117. In this rule, if the factor of 
safety be taken at 20, the result will be sufficiently large to 
allow for necessary cuttings at the joints. Therefore, if the 
beam be of Georgia pine, equation (16.), Art. 117, becomes — 

. 27000 X 20 - 

^=^7655^- =341; 

or, say, 35 inches. This is ample to resist the tensile strain ; 
but, to resist the transverse strains to which such a long 
piece of timber is subjected in the hands of the workman, 
it would be proper to make it, say, 6x9. 



STRAIN UPON THE RAFTER. 205 

227. — The Rafter. — A rafter, like a post, is subject to a 
compressive force, and is liable to fail in three ways, name- 
ly : by flexure, by being crushed, or by crushing the material 
against which it presses. To render it entirely safe, there- 
fore, it is requisite to ascertain the requirements for resisting 
failure in each of these three ways. 

Of these it will be convenient to consider, first, that of 
the Kability to being crushed. The rule for this is found in 
Art, 107. Let the rafter be of Georgia pine, then the value 
of C, Table I., will be 9500. The strain in the rafter {Art. 
225) is 31,200 pounds. Now, taking the value of a, the fac- 
tor of safety, at lo, we have, by Rule VI. {Art. 107.) — 

31200 X 10 

A — • = 32-737; 

9500 J /J/' 

or, 33 inches area of cross-section. This is the size of the 
rafter at its smallest section ; for example, at any one of the 
joints where it is customary to reduce the area by cutting 
for the struts and rods. 

Again : Let the liability of the rafter to flexure be now 
considered. For this we have a rule in Art. 114. The 
length of the rafter between unsupported points is nearly 9! 
feet, or 9f X 12 = 1 16 inches. Let the thickness of the rafter 
be taken at 6 inches. Then, by Rule ILX. {Art. 114), we 
have — 



h — ^^^(^ + 1^^') _ 31200 X 10(1+ f X .00109 X r') 
C t ~ 9500 X 6 

/ 116 , 

^=7==-5-=-i9i; 19*' = 373-8. 

Then, f x .00109 x 373-8 = 0-611127 
adding unity = i • 

I .611127 
Substituting this, we have — 

_ 31200 X IPX 1 .611127 _ 502671 .624 

^ ~ 9500 x"6 ~~^o^^~"'"^'^^9; 



206 CONSTRUCTION. 

or, to resist flexure the breadth is required to be 8-82, or, 
say, 9 inches ; or, the rafter is to be 6 xq inches at the foot. 
The strain in the rafter at the upper end is only half that at 
the foot ; the area of cross-section, therefore, at the head 
need not be more than half that which is required at the 
foot ; but it is usual to make it there about f of the size at the 
foot. In this case it would be, therefore, 6x6 inches at the 
upper end. 

Lastly, the requirement to resist crushing the surfaces 
against which the rafter presses is to be considered. 

The fibres of timber yield much more readily when 
pressed together by a force acting at ri£-/it angles to the di- 
rection of their length than when it acts in a line with their 
length. 

The value of timber subjected to pressure in these two 

ways is shown in Arts. 94, 98. In Table I., the value per 

square inch of the first stated resistance is expressed by P, 

C 
and the ultimate resistance of the other by — . The value 

■^ a 

of timber per square inch to safely resist crushing may be ex- 

C 
pressed by — , in which a is the factor of safety. Timber 

pressed in an oblique direction will resist a force exceeding 

C 
that expressed by P, and less than that expressed by — . 

When the angle of inclination at which the force acts is just 

C 

45°, then the force will be an average between P and — . 

And for any angle of inclination, the force will vary inverse- 
ly as the angle ; approaching P as the angle is enlarged, 

C 
but approaching — as the angle is diminished. It will be 

C 
equal to — when the angle becomes zero, and equal P 

when the angle becomes 90°. The resistance of timber per 
square inch to an oblique force is therefore expressed by — 



90 



(5-P); (102.) 



RESISTANCE OF SURFACES. 20/ 

where ^° equals the complement of the angle of inclination. 
In a roof, ^° is the acute angle formed by the rafter with 
a vertical line. If no convenient instrument be at hand to 
measure the angle, describe an arc upon the plan of the 
truss — thus : with C B {Fig. 84) for radius, describe the arc 
B g, and get the length of this arc in feet by stepping it off 
with a pair of dividers. Then — 

-- = 0-631 f; 

90 // 

where k equals the length of the arc, and h equals B Cy the 
height of the roof. Therefore — 

k fC 



il/=:P+o.63||(^--p) (103.) 



equals the value of timber per square inch in a tie-beam, C 
and P being obtained from Table I., Art. 94. When C for 
the kind of wood in the tie-beam exceeds C set opposite the 
kind of wood in the rafter, then the latter is to be used in 
the rules instead of the former. 

The value of M, equation (103.), is the resistance per 
square inch of the surface pressed at the foot of the rafter. 
The resistance of the entire surface will therefore he MA, 
where A equals the area of the joint. Then, when the re- 
sistance equals the strain, we will have — 

MA=S = A[p+o.6i%l{^-p) 

from which we have — 

A 



^-^^•^3f|g-^' 



(104.) 



in which 5 is the strain to be resisted. 

Now, the end of the rafter must be of sufficient size to 
afford a joint the area of which will not be less than that 
expressed by A in equation (104.). 

For example, the strain to which the rafter, Fig. 84, is 
subject at its foot is ascertained to be(^r/. 225) 31,200 pounds. 
For Georgia pine, the material of the tie-beam, P — goo 
{Art. 94, Table /.), and (7 = 9500. 



208 CONSTRUCTION. 

The length of the arc Bo- is about 14-4 feet; the height 
B Cis 13 feet. Let a, the factor of safety, be taken at 10, 
then we have (104.) — 



900 + (o-63t X -W ) (910 _ 900) 

_ 3T200 _ 

~ 900 + (O • 705 X 50) ~ ^'^ * ^ ' 

or, the superficial area of the bearing at the joint required 
to prevent crushing the tie-beam is 33^^ inches. 

The results of the computations show that the rafter is 
required to be 6 inches thick, 9 inches wide at the foot, and 
6 inches wide at the top. It is also ascertained that, in cut- 
ting for the bearing for the struts and boring for the sus- 
pension-rods, it is required that there shall be at least 33 
inches area of cross-section left intact ; and, farther, that the 
area of the surface of the joint against the tie-beam should 
not be less than 33^ inches. 

228. — The Braces. — Each brace is subject to compres- 
sion, and is liable to fail if too small, in the same manner as the 
rafter. Its size is to be ascertained, therefore, in the manner 
described for the rafter ; which need not be here repeated, 
except, perhaps, as to the liability to fail by flexure ; for in this 
case we have the breadth given, and need to find the thick- 
ness. The breadth of the- brace is fixed b}^ the thickness of 
the rafter, for it is usual to have the tw^o pieces flush with 
each other. Rule XI. {ArL 114) is to be used, but with this 
difference, namely : instead of the thickness, use the breadth 
as one of the factors in the divisor. Thus — 

t = '-.-^ -^. (105.) 

In working this rule, it is required, in order to get the 
value of r, the ratio between the height and thickness, to 
assume the thickness before it is ascertained ; and after com- 
putation, if the result shows that the assumed value was not 
a near approximation, a second trial will have to be made. 
Usually the first trial will be sufficient. 



STRAIN UPON BRACES. 209 

For example, the brace D E \s about 9I feet or 1 16 inches 
long. As the strain in it is only 5200 pounds, the thickness 
will probably be not over 3 inches. Assuming it at this, we 

/ 

t 
Therefore, we have — 



have r=-= J-p = 38I ; the square of which is about 1495. 



I X 0-00109 X 1495 = 2-4445 
add unity = i. 

3-4445 
The equation reduces, therefore, to this — 

_ 5200x10x3 -4445 _ ^ 

^- 5^^^^6 -3-1424, 

or, the required thickness of the brace is 34- inches, or the 
brace should be, say, 3J x 6 inches. In this case the result is 
so near the assumed value, a second trial is not needed. 

For the second brace, we have the length equal to about 
12J feet or 147 inches ; and the strain equal to 6575 pounds 
{Art. 225). The ratio, therefore, may be obtained by assum- 
ing the thickness, say, at 4. With this, we have — 

/ 
r =- =i|^= 36-75 ; the square of which is I350t*^. 

With this value of r" — 

I X -00109 X i350y^^ = 2-2081 
add unity = i. 



Then- 



3 -2081 
6575x10x3.2081 

^^55^ =3-7006. 



Comparing this result with the assumed value of t = 4, 

we find the difference so great as to require a second trial. 

As the value of r was taken too low, the result obtained is 

correspondingly low. The true value is somewhere between 

3 • 7 and 4. Assume it now, say, at 3-9. With this value, we 

have — 

/ 147 
^ = ^= —- = 37-692 ; the square of which is 1420-7. 



210 CONSTRUCTION. 

With this value of r'— 

f X -00109 X 1420-7 = 2-32282 
add unity =: i • 

3-32282 



Then — 



_6575 X 10 X 3-32282 
^~ 95"6o76 =3-833. 



This result is a trifle less than the assumed value, 3-9. The 
true value is between these, and probably is about 3-86. 
This is quite near enough for use. This brace, therefore, is 
required to be 3 -86 x 6 inches, or, say, 4x6 inches. 

229. — The Suspen§ion-Rod§.— These are usually made of 
wrought iron. This metal, when of excellent quality, may 
be safely trusted with 12,000 pounds per inch sectional area. 
But it is usual, for good work, to compute the area at only 
9000 pounds per inch, and, as ordinarily made, these rods 
ought not to be loaded with more than 7000 pounds. The 
strain divided by this value per inch of the metal will give 
the sectional area of cross-section. For example, the strain in 
the rod jD G, Fig, 84, is 867 pounds {Art, 225); therefore — 

A 867 

A — — ^=:0-i24; 
7000 ^ 

or, the sectional area required is only an eighth of an inch. 
By reference to the table of areas of circles in the Appen- 
dix, the diameter of a rod containing the required area, as 
above, will be found to be a little less than half an inch. A 
rod half an inch in diameter will therefore be of .ample 
strength. For appearance's sake, however, no rod in a truss 
should be less than | of an inch in diameter. 

The rod FE has to resist a strain of 3192 pounds. For 
this, then, we have — 

^=^^-^ = 0.456. 
7000 ^^ 

A reference to the table of areas shows that a rod contain- 



ROOF-BEAMS. 211 

ing this area would be a little more than f of an inch in di- 
ameter ; it would be of ample strength, say, at ^ of an inch 
in diameter. 

The rod C B, at the centre, has to carry a strain of 10,167 
pounds. For this, then, we have — 

10167 

A — — 1-452. 

7000 ^^ 

A reference to the table of areas shows that this rod should 
be i^ inches in diameter. 

230. — Roof-Beams, Jack-RaHers, and Purlins. — These 
timbers are subject to loads nearly uniformly distributed, 
and their dimensions may be obtained by Rule XXX., equa- 
tion {i^'.), Art. 140. In this equation, U — cf I {Art. 152). 
Substituting this value for U, and r I for (J, equation (35.) be- 
comes — 

\'K)Fr 
and putting for r the rate of deflection, .04, we have — 

bd^ — -" ^ ^ , (106.) 

a formula convenient for roof-timbers. 

Example. — In a roof where the roofing is to be supported 
on white-pine roof-beams 10 feet long, placed 2\ feet from 
centres, and where the load per foot superficial is to be 40 
pounds, including wind and snow : Avhat should be the di- 
mensions of the roof-beams? By equation (106.) — 

, ,, 40 X 2^- X 10^ 

bd'=.^—T-- =538.8. 

0.064x2900 ■'^ 

Now if b, the breadth, be fixed, say, at 3, then — 

.= = ^^=.796; 

^=: 5-64 nearly. 



212 



CONSTRUCTION. 



The roof-beams, therefore, require to be 3 x 5I, or, say, 3x6. 
All pieces of timber subject to cross-strains will sustain 
safely much greater strains when extended in one piece over 
two, three, or more distances between bearings ; therefore, 
roof-beams, jack-rafters, and purlins should, if possible, be 
made in as long lengths as practicable ; the roof-beams and 
purlins laid on, not framed into, the principal rafters, and 
extended over at least two spaces, the joints alternating on 
the trusses ; and likewise the jack-rafters laid on the purlins 
in long lengths. 

231. — Five Examples of Roof^: are shown at Figs. 85, 86, 
87, 88, and 89. In Fig. 85, a is an iron suspension-rod, b, b are 

braces. In Fig. 86, a, a, and 
b are iron rods, and d, d, c, c 
are braces. In Fig. Zj, a, b 
are iron rods, d, d braces, and 
c the straining beam. In 
Fig. 85. Fig. 88, a, a, b, b are iron rods, 

e, e, d, d are braces, and r is a straining beam. In Fig. 89, pur- 





lins are located at P P, etc. ; the inclined beam that lies upon 
them is the jack-rafter ; the post at the ridge is the king- 



TRUSS WITH BUILT-RIB. 



213 



post, the others are queen-posts. In this design the tie-beam 
is increased in height along the middle by a strengthening 
piece {Art. 163), for the purpose of sustaining additional 
weight placed in the room form- 
ed in the truss {Art. 216). 

Fi£: 90 shows a method of 
constructing a truss having a 
built-rib in the place of prin- 
cipal rafters. The proper form 
for the curve is that of the par- 
abola {Art. 560). This curve, 
when as fiat as is described in 
the figure, approximates so close- 
ly to that of the circle that the 
latter may be used in its stead. 
The height, a b^ is just half of 
a c, the curve to pass through 
the middle of the rib. The rib 
is composed of two series of 
abutting pieces, bolted together. 
These pieces should be as long 
as the dimensions of the timber 
will admit, in order that there 
may be but few joints. The sus- 
pending pieces are in halves, 
notched and bolted to the tie- 
beam and rib, and a purlin is 
framed upon the upper end of 
each. A truss of this construc- 
tion needs, for ordinary roofs, 
no diagonal braces between the 
suspending pieces, but if extra 
strength is required the braces 
may be added. The best place 
for the suspending pieces is at 
the joints of the rib. A rib of this kind will be sufficiently 
strong if the area of its section contain about one fourth 
more timber than is required for that of a rafter for a roof 
of the same size. The proportion of the depth to the thick- 
ness should be about as 10 to 7. 




214 



CONSTRUCTION. 



232. — Roof-Truss witli Elevated Tie-Beam. — Designs 
such as are shown in Fig. 91 have the tie elevated for the ac- 
commodation of an arch in the ceiUng. This and all similar 
designs are seriously objectionable, and should always be 




avoided ; as the smaii height gained by the omission of the 
tie-beam can never compensate for the powerful lateral 
strains which are exerted by the oblique position of the 




supports, tending to separate the walls. Where an arch is 
required in the ceiling, the best plan is to carry up the 
walls as high as the top of the arch. Then, by using a 
horizontal tie-beam, the oblique strains will be entirely re- 



HIP-ROOFS. 



215 



moved. It is well known that many a public building has 
been all but ruined by the settling of the roof, consequent 
upon a defective plan in the formation of the truss in this 
respect. It is very necessary, therefore, that the horizontal 




Fig. 91. 



tie-beam be used, except where the walls are made so strong 
and firm by buttresses, or other support, as to prevent a 
possibility of their separating. (See Art. 212.) 




233. — Hip-Roofs: Lines and BeTil§. — The lines a b and 
be, in Fig. 92, represent the walls at the angle of a building; 
b c\s> the seat of the hip-rafter, and g f oi a jack or cripple 
rafter. Draw e h at right angles to be, and make it equal 



2l6 



CONSTRUCTION. 



to the rise of the roof ; join b and h, and h b will be the 
length of the hip-rafter. Through e draw di at right angles 
to <^^; upon b, with the radius bh, describe the arc hi^ 
cutting di in i\ join b and /, and extend gfto meet b i iny; 
then gj will be the length of the jack-rafter. The length 
of each jack-rafter is found in the same manner — by extend- 
ing its seat to cut the line b i. From / draw fk at right 
angles to fg, also// at right angles to be-, make/y^ equal 
to// by the arc Ik, or make g k equal to gj by the arc j k ; 
then the angle at / will be the top-bevil of the jack-rafters, 
and the one at k will be the down-bevil.^ 

234. — Tlie Backings of the Hip-Rafter. — At any con- 
venient place in <^ ^ {Fig. 92), as 0, draw m n at right angles to 
be) from 0, tangical to b h, describe a semicircle, cutting b e 
in s ; join m and s and n and s\ then these lines will form at 
s the proper angle for bevilling the top of the hip-rafter. 

DOMES.f 

235. — Domes. — The usual form for domes is that of the 
sphere ; the base circular. When the interior dome does not 




Fig. 93. 



rise too high, a horizontal tie may be thrown across, by 
which any degree of strength required may be obtained. 



* The lengths and bevils of rafters for rooUva/leys can also be found by the 
above process. 

f See also Art 68. 



CONSTRUCTION OF DOMES. 



217 



Fig. 93 shows a section, and Fig. 94 the plan, of a dome of 
this kind, a b being the tie-beam in both. Two trusses of 
this kind {Fig. 93), parallel to each other, are to be placed 
one on each side of the opening in the top of the dome. 
Upon these the whole framework is to depend for support, 




Fig. 94. 



and their strength must be calculated accordingly. (See 
Arts, yo to 80 and 214 to 222.) If the dome is large and of 
importance, two other trusses may be introduced at right 
angles to the foregoing, the tie-beams being preserved in 




Fig. 95. 

one continuous length by framing them high enough to pass 
over the others. 



236. — Ribbed Dome. — When the interior must be kept 
free, then the framing may be composed of a succession of 
ribs standing upon a continuous circular curb of timber, as 



2l8 



CONSTRUCTION. 



seen at Figs. 95 and 96 — the latter being a plan and the former 
a section. This curb must be well secured, as it serves in 
the place of a tie-beam to resist the lateral thrust of the ribs. 
In small domes these ribs may be easily cut from wide 
plank ; but where an extensive structure is required, they 
must be built in two thicknesses so as to break joints^ in the 
same manner as is described for a roof at Art. 231. They 
should be placed at about two feet apart at the base, and 
strutted as at a in Fig. 95. 




Fig. 96. 

The scantling of each thickness of the rib may be as fol- 
lows : 

For domes of 24 feet diameter, i x 8 inches, 

a u u 26 '' '' i^x 10 '' 

U ^Q u u 2 ^ j^ 

" 90 '' '' 2ix 13 " 

u jQg u u 2 ^ j^ 



237. — Dome : Curve of Equilibrium. — The surfaces of a 
dome may be finished to any curve that may be desired, but 
the framing should be constructed of such form that the 
curve of cqiiilibriiini shall be sure to pass through the middle 
of the depth of the framing. The nature of this curve is 
such that, if an arch or dome be constructed in accordance 
with it, no one part of the structure will be less capable than 
another of resisting the strains and pressures to which the 
whole fabric may be exposed. The curve of equilibrium for 
an arched vault or a roof, w^here the load is equally diffused 



CURVE OF EQUILIBRIUM. 



219 



over the whole surface, is that of a parabola (ArL46o); for 
a dome having no lantern, tower, or cupola above it, a cubic 
parabola {Fig. 97) ; and for one having a tower, etc., above it, 
a curve approaching that of an hyperbola must be adopted, 
as the greatest strength is required at its upper parts. If 
the curve of a dome be circular (as in the vertical section, 
Fig,g^), the pressure will have a tendency to burst the dome 
outwards at about one third of its height. Therefore, when 
this form is used in the construction of an extensive dome, 
an iron band should be placed around the framework at 
that height ; and whatever may be the form of the curve, a 
band or tie of some kind is necessary around or across the 
base. 



J^""^ 


'^ 


oy^ 




X 


y \ \ 


"A i 1 


/[ 1 ' 


/ ' ' 





<^ J i h g / e d 

Fig. 97. 



If the framing be of a form less convex than the curve of 
equilibrium, the weight will have a tendency to crush the 
ribs inwards, but this pressure may be effectually overcome 
by strutting between the ribs ; and hence it is important 
that the struts be so placed as to form continuous horizontal 
circles. 



238. — Cubic Parabola Computed. — Let a b {Fig. 97) be 
the base, and b c the height. Bisect a b at d, and divide a d 
into 100 equal parts ; of these give dc 26, cf i8J,/^ hJ,^/^ 
12J-, h i lof, ij 9I and the balance, 8|, Vo j a ; divide b c into 
8 equal parts, and from the points of division draw lines 
parallel to a /^ to meet perpendiculars from the several points 



220 



CONSTRUCTION. 



of division in a b, at the points o, o, o, etc. Then a curve 
traced through these points will be the one required. 

239. — Small Domes over Stairways : are frequently made 
elliptical in both plan and section ; and as no two of the ribs 




in one quarter of the dome are alike in form, a method for 
obtaining the curves may be useful. 




Fig. 99. 

To find the curves for the ribs of an elliptical dome, let 
abed {Fig. 98) be the plan of a dome, and ef the seat of 
one of the ribs. Then take e f ior the transverse axis and 
twice the rise, og, of the dome for the conjugate, and de- 



COVERING OF DOMES. 



221 



scribe (according to Arts. 548, 549, etc.) the semi-ellipse 
eg-/, which will be the curve required for the rib eg-/. The 
other ribs are found in the same manner. 

24-0. — Covering for a Spherical Dome. — To find the 
shape, let A [Fig. 99) be the plan, and B the section, of a given 
dome. From a draw ^ ^ at right angles \.o ab \ find the 
stretch-out {Art. 524) of ob, and make dc equal to it; divide 
the arc b and the line d c each into a like number of equal 
parts, as 5 (a large number will insure greater accuracy than 
a small one) ; upon c, through the several points of division 
in cd, describe the arcs do, i e i, 2/2, etc. ; make do equal 
to half the width of one of the boards, and draw os parallel 




to ^ ^ ; join s and a, and from the points of division in the arc 
ob drop perpendiculars, meeting asintjk/; from these 
points draw ^ 4,7*3, etc., parallel to <^^; make do,e i, etc., on 
the lower side of ac, equal to do, e i, etc., on the upper side ; 
trace a curve through the points 0, 1,2, 3, 4, c, on each side 
of dc ; then ^ <; ^ will be the proper shape for the board. By 
dividing the circumference of the base A into equal parts, 
and making the bottom, do, of the board of a size equal to 
one of those parts, every board may be made of the same 
size. In the same manner as the above, the shape of the 
covering for sections of another form may be found, such as 
an ogee, cove, etc. 

To find the curve of the boards when laid in horizontal 
courses, \^t A B C {Fig. 100) be the section of a given dome, 



222 



CONSTRUCTION. 



and Z>^ its axis. Divide BC into as many parts as there 
are to be courses of boards, in the points i, 2, 3, etc. ; through 
I and 2 draw a line to meet the axis extended at a ; then a 
will be the centre for describing the edges of the board F. 
Through 3 and 2 draw 3 b ; then b will be the centre for de- 




FlG. lOI, 

scribing F. Through 4 and 3 draw A,d\ then ^ will be the 
centre for G. B is the centre for the arc i 0. If this method 
is taken to find the centres for the boards at the base of the 
dome, they would occur so distant as to make it impracti- 
cable ; the following method is preferable for this purpose : 
G being the last board obtained by the above method, ex- 
tend the curve of its inner edge until it meets the axis, D B, 




in e ; from 3, through e, draw 3 /, meeting the arc y^ ^ in /; 
join / and 4, / and 5 , and /and 6, cutting the axis, D B, in s, n, 
and in ; from 4, 5, and 6 draw lines parallel to A (7 and cutting 
the axis in c, p, and r ; make c 4 {jFig. loi) equal to ^ 4 in the pre- 
vious figure, and cs equal to c s also in the previous figure ; 
then describe the inner edge of the board //", according to 
Art. 516; the outer edge can be obtained by gauging from 
the inner edge. In like manner proceed to obtain the next 



DESIGNS OF BRIDGES. 



223 



board — taking/ 5 for. half the chord, and pn for the height 
of the segment. Should the segment be too large to be de- 
scribed easily, reduce it by finding intermediate points in the 
curve, as at ^r/. 515. 

24 L — Polygonal Dome: Form of Angle-Rib. — To ob- 
tain the shape of this rib, let A G H {Fig. 102) be the plan of 
a given dome, and C D ^ vertical section taken at the line 
ef. From i, 2, 3, etc., in the arc CD draw ordinates, paral- 
lel to A D, to meet fG ; from the points of intersection on 
fG draw ordinates at right angles to f G \ make ^ i equal 
to I, s 2 equal to 2, etc. ; then GfB, obtained in this way, 
will be the angle-rib required. The best position for the 
sheathing-boards for a dome of this kind is horizontal, but if 
they are required to be bent from the base to the vertex, 
their shape may be found in a similar manner to that shown 
at Fig. 99. 

BRIDGES. 




Fig. 103 



242. — Bridges. — Of plans for the construction of bridges, 
perhaps the following are the most useful. Fig. 103 shows a 
method of constructing wooden bridges where the banks 
of the river are high enough to permit the use of the tie- 
beam, ab. The upright pieces, c d, are notched and bolted 
on in pairs, for the support of the tie-beam. A bridge ot 
this construction exerts no lateral pressure upon the abut- 
ments. This method may be employed even where the banks 
of the river are low, by letting the timbers for the roadway 
rest immediately upon the tie-beam. In this case the irame- 
work above will serve the purpose of a railing. 



224 



CONSTRUCTION. 



Fig. 104 exhibits a wooden bridge without a tie-beam. 
Where staunch buttresses can be obtained this method may 
be recommended ; but if there is any doubt ot their stabihty, 
it should not be attempted, as it is evident that such a sys- 
tem of framing is capable of a tremendous lateral thrust. 




Fig. 104. 



24-3. — Bridges: Built-Rib. — Fig. 105 represents a bridge 
with a built-rib (see Art. 231) as a chief support. The curve 
of equihbrium will not differ much from that of a parabola ; 
this, therefore, may be used — especially if the rib is made 




Fig. 105. 

gradually a little stronger as it approaches the buttresses. 
As it is desirable that a bridge be kept low, the following 
table is given to show the least rise that may be given to the 
rib. 



Span in Feet. 


Least Rise in Feet 


Span in Feet. 


Least Rise in Feet 


Span in Feet. 


Least Rise in Feet 


30 


0-5 


120 


7 


280 


24 


40 


0-8 


140 


8 


300 


28 


50 


1-4 


160 


10 


320 


32 


60 


2 


180 


II 


350 


39 


70 


2* 


200 


12 


380 


47 


80 


3 


220 


14 


400 


53 


90 


4 


240 


17 






TOO 


5 


260 


20 







DIMENSIONS OF THE BUILT-RIB. 22'5 

The rise should never be made less than this, but in all 
cases greater if practicable ; as a small rise requires a greater 
quantity of timber to make the bridge equally strong. The 
greatest uniform weight with which a bridge is likely to be 
loaded is, probably, that of a dense crowd of people. This 
may be estimated at 70 pounds per square foot, and the fram- 
ing and gravelled roadway at 230 pounds more ; which 
amounts to 300 pounds on a square foot. The following 
rule, based upon this estimate, may be useful in determining 
the area of the ribs. 

Rule LXVI I.— Multiply the width of the bridge by the 
square of half the span, both in feet, and divide this pro- 
duct by the rise in feet multiplied by the number of ribs ; 
the quotient multiplied by the decimal o-ooii will give the 
area of each rib in feet. When the roadway is only planked, 
use the decimal 0-0007 instead of o-ooii. 

Example. — What should be the area of the ribs for a 
bridge of 200 feet span, to rise 15 feet and be 30 feet wide, 
with three curved ribs? The half of the span is 100, and 
its square is loooo ; this multiplied by 30 gives 300000, and 
15 multiplied by 3 gives 45; then 300000 divided by 45 
gives 6666|, which multiplied by o-ooii gives 7-333 feet or 
1056 inches for the area of each rib. Such a rib may be 24 
inches thick by 44 inches deep, and composed of 6 pieces, 
2 in width and 3 in depth. 

The above rule gives the area of a rib that would be 
requisite to support the greatest possible uniform load. 
But in large bridges, a variable load, such as a heavy wagon, 
is capable of exerting much greater strains ; in such cases, 
therefore, the rib should be made larger.^* 

In constructing these ribs, if the span be not over 50 feet, 
each rib may be made in two or three thicknesses of timber 
(three thicknesses is preferable), of convenient lengths bolted 
together; but in larger spans, where the rib will be such as 
to render it difficult to procure timber of sufficient breadth, 
they may be constructed by bending the pieces to the proper 
curve and bolting them together. In this case, where tim- 

* See Tred^old's Carpentry by Hurst, Arts. 174 to 177. 



226 



CONSTRUCTION. 



ber of sufficient length to span the opening cannot be ob- 
tained, and scarfing is necessary, such joints naust be made 
as will resist both tension and compression (see Fig. 1 14). 
To ascertain the greatest depth for the pieces which compose 
the rib, so that the process of bending may not injure their 
elasticity, multiply the radius of curvature in feet by the 
decimal 0-05, and the product will be the depth in inches. 
£;r(^;;2//^.— Suppose the curve of the rib to be described 
with a radius of 100 feet, then what should be the depth? The 
radius in feet, 100, multiplied by 0-05 gives a product of 5 
inches. White pine or oak timber 5 inches thick would 
freely bend to the above curve ; and if the required depth 
of such a rib be 20 inches, it would have to be composed of at 
least 4 pieces. Pitch pine is not quite so elastic as white 
pine or oak— its thickness may be found by using the deci- 
mal 0-046 instead of 0-05. 




Fig. 106. 



244-- — Bridges: Framed Rib. — In spans of over 250 
feet, "d. framed rib, as in Fig. 106, would be preferable to the 
foregoing. Of this, the upper and the lower edges are 
formed as just described, by bending the timber to the proper 
curve. The pieces that tend to the centre of the curve, 
called radials,.7XYQ notched and bolted on in pairs, and the 
cross-braces are halved together in the middle, and abut end 
to end between the radials. The distance between the ribs 
of a bridge should not exceed about 8 feet. The roadway 
should be supported by vertical standards bolted to the ribs 



THE ROADWAY AND ABUTiMENTS. 22/ 

at about every lo to 15 feet. At the place where they rest 
on the ribs, a double, horizontal tie should be notched and 
bolted on the back of the ribs, and also another on the un- 
derside ; and diagonal braces should be framed between the 
standards, over the space between the ribs, to prevent lat- 
eral motion. The timbers for the roadway may be as lig-ht 
as their situation will admit, as all useless timber is only an 
unnecessary load upon the arch. 

245. — Bridges: Roadway. — If a roadway be 18 feet 
wide, tv/o carriages can pass without inconvenience. Its 
width, therefore, should be either 9, 18, 27, or 36 feet, ac- 
cording- to the amount of travel. The width of the foot-> 
path should be two feet for every person. When a stream 
of water has a rapid current, as few piers as practicable 
should be allowed to obstruct its course ; otherwise the 
bridge will be liable to be swept away by freshets. When 
the span is not over 300 feet, and the banks of the river are 
of sufficient height to admit of it, only one arch should be 
employed. The rise of the arch is limited by the form of 
the roadwa^v, and by the height of the banks of the river 
(see Art. 243). The rise of the roadway should not exceed 
one in 24 feet, but as the framing settles about one in 72, the 
roadway should be framed to rise one in 18, that it may be 
one in 24 after settling. The commencement of the arch at 
the abutments — the spring, as it is termed — should not be* 
below high-water mark ; and the bridge should be placed at 
right angles with the course of the current. 

246. — Bridges: Abiitmeiats. — The best material for the 
abutments and piers of a bridge is stone ; and no other 
should be used. The following rule is to determine the ex- 
tent of the abutments, they being rectangular, and built with 
stone weighing 120 pounds to a cubic foot. 

Rule LXVHI— Multiply the square of the height of the- 
abutment by 160, and divide this product by the weight of a 
square foot of the arch, and by the rise of the arch ; add 
unity to the quotient, and extract the square root. Dimin- 
ish the square root by unity, and multiply the root sj dimin- 



228 CONSTRUCTION. 

ished by half the span of the arch, and by the weight of a 
square foot of the arch. Divide the last product by 120 
times the height of the abutment, and the quotient will be 
the thickness of the abutment. 

Example. — Let the height of the abutment from the base 
to the springing of the arch be 20 feet, half the span 100 feet, 
the weight of a square foot of the arch, including the great- 
est possible load upon it, 300 pounds, and the rise of the arch 
18 feet: what should be its thickness? The square of the 
height of the abutment, 400, multiplied by 160 gives 64000, 
and 300 by 18 gives 5400; 64000 divided by 5400 gives a 
quotient of 11-852; one added to this makes 12-852, the 
square root of which is 3-6 ; this, less one is 2-6 ; this mul- 
tiplied by 100 gives 260, and this again by 300 gives 78000; 
this divided by 120 times the height of the abutment, 2400, 
gives 32 feet 6 inches, the thickness required. 

. The dimensions of a pier will be found by the same rule ; 
for, although the thrust of an arch may be balanced by an 
adjoining arch when the bridge is finished, and while it re- 
mains uninjured, yet, daring the erection, and in the event 
of one arch being destroyed, the pier should be capable of 
sustaining the entire thrust of the other. 

Piers are sometimes constructed of timber their princi- 
pal strength depending on piles driven into the earth ; but 
such piers should never be adopted where it is possible to 
avoid them ; for, being alternately wet and dry, they decay 
much sooner than the upper parts of the bridge. Spruce 
and elm are considered good for piles. Where the height 
from the bottom of the river to the roadway is great, it is a 
good plan to cut them off at a little below low-water mark, 
cap them with a horizontal tie, and upon this erect the posts 
for the support of the roadway. This method cuts off the 
pan ihat Is continually wet from that which is only occa- 
sionally so, and thus affords an opportunity for replacing the 
upper part. The pieces which are immersed will last a 
great length of time, especially when of elm ; for it is a 
Well-established fact that timber is less durable when subject 
to alternate dryness and moisture than when it is either con- 
tinually Avet or continually dry. It has been ascertained that 



CENTRING FOR BRIDGES. 



229 



the piles under London Bridge, after having been driven 
about 600 years, were not materially decayed. These piles 
are chiefly of elm, and wholly immersed. ; 

247. — Centres for Stone Bridges. — Fig. 107 is a design 
for a centre for a stone bridge where intermediate supports, 
as piles driven into the bed of the river, are practicable. Its 
timbers are so distributed as to sustain the weight of the 
arch-stones as they are being laid, without destroying the 
original form of the centre ; and also to prevent its destruc- 
tion or settlement, should any of the piles be swept away. 
The most usual error in badly-constructed centres is that 
the timbers are disposed so as to cause the framing to 
rise at the crown during the laying of the arch-stones up 




the sides. To remedy this evil, some have loaded the crown 
with heavy stones ; but a centre properly constructed will 
need no such precaution. i 

Experiments have shown that an arch-stone does not press 
upon the centring until its bed is inclined to the horizon at 
an angle of from 30 to 45 degrees, according to the hardness 
of the stone, and whether it is laid in mortar or not. For 
general purposes, the point at which the pressure com- 
mences may be considered to be at that joint which forms 
an angle of 32 degrees with the horizon. At this point the 
pressure is inconsiderable, but gradually increases towards 
the crown. The following table gives the portion of the 
weight of the arch-stones that presses upon the framing at 
the various angles of inclination formed by the bed of the 



230 



CONSTRUCTION. 



Stone with the horizon. The pressure perpendicular to the 
curve is equal to the weight of the arch-stone multiplied by 
the decimal — 

•o, when the angle of mclination is 32 degrees. 



04 






34 




08 






36 




12' 






38 




17 






40 




21 ' * 






42 




25 






44 




29 






46 




33 






48 




37 






50 




4 






52 




44 






54 




48 






56 




52 






58 




54 






60 






Fig. 108. 



From this it is seen that at the inclination of 44 degrees the 
(pressure equals one quarter the weight of the stone ; at 57 
degrees, half the weight; and when a vertical line, as ad 
{Fig. 108), passing through the centre of 
gravity of the arch-stone, does not fall 
within its bed, c d, the pressure may be con- 
sidered equal to the whole weight of the 
stone. This will be the case at about 60 
degrees, when the depth of the stone is 
double its breadth. The direction of these 
pressures is considered in a line with the radius of the curve. 
The weight upon a centre being known, the pressure may be 
estimated and the timber calculated accordingly. But it 
must be remembered that the whole weight is never placed 
upon the framing at once — as seems to have been the idea 
had in view by the designers of some centres. In building 
the arch, it should be commenced at each buttress at the 
same time (as is generally the case), and each side should 
progress equally towards the crown. In designing the fram- 



CENTRE FOR A STONE BRIDGE. 



^51 



ing, the effect produced by each successive layer of stone 
should be considered. The pressure of the stones upon one 
side should, by the arrangement of the struts, be counter- 
poised by that of the stones upon the other side. 

Over a river whose stream is rapid, or where it is neces- 
sary to preserve an uninterrupted passage for the purposes 
of navigation, the centre must be constructed without in- 
termediate supports, and without a continued horizontal tie 
at the base ; such a centre is shown at Fig. 109. In laying 
the stones from the base up to a and c, the pieces bd and 
bd act as ties to prevent an}^ rising at b. After this, while 
the stones are being laid from a and from c to b, they act as 
struts; the piece f g is added for additional security. 
Upon this plan, with some variation to suit circumstances, 




Fig. 109. 

centres may be constructed for any span usual in stone- 
bridge building. 

In bridge centres, the principal timbers should abut, and 
not be intercepted by a suspension or radial piece between. 
These should be in halves, notched on each side and bolted. 
The timbers should intersect as little as possible, for the 
more joints the greater is the settling ; and halving them 
together is a bad practice, as it destroys nearly one half the 
strength of the timber. Ties should be introduced across, 
especially where many timbers meet ; and as the centre is 
to serve but a temporary purpose, the whole should be de- 
signed with a view to employ the timber afterwards for 
other uses. For this reason, all unnecessary cutting should 
be avoided. 



232 CONSTRUCTION. 

Centres should be sufficiently strong to preserve a 
staunch and steady form during the whole process of build- 
ing; for any shaking or trembling will have a tendency to 
prevent the mortar or cement from setting. For this pur- 
pose, also, the centre should be lowered a trifle immedi- 
ately after the key-stone is laid, in order that the stones may 
take their bearing before the mortar is set ; otherwise the 
joints will open on the underside. The trusses, in centring, 
are placed at the distance of from 4 to 6 feet apart, accord- 
ing to their strength and the weight of the arch. Between 
every two trusses diagonal braces should be introduced to 
prevent lateral motion. 

In order that the centre may be easily lowered, the 
frames, or trusses, should be placed upon wedge-formed 
sills, as is shown at d {Fig. 109). These are contrived so as 
to admit of the settling of the frame by driving the wedge 
d with a maul, or, in large centres, with a piece of timber 
mounted as a battering-ram. The operation of lowering a 
centre should be very slowly performed, in order that the 
parts of the arch may take their bearing uniformly. The 
wedge pieces, instead of being placed parallel with the 
truss, are sometimes made sufficiently long and laid through 
the arch, in a direction at right angles to that shown at Fig. 
109. This method obviates the necessity of stationing men 
beneath the arch during the process of lowering ; and was 
originally adopted with success soon after the occurrence of 
an accident, in lowering a centre, by which nine men were 
killed. 

To give some idea of the manner of estimating the pres- 
sures, in order to select timber of the proper scantling, cal- 
culate the pressure {Art. 247) ot the arch-stones from i to b 
(Fig. 109), and suppose half this pressure concentrated at a, 
and acting in the direction a f. Then, by the parallelogram 
of forces {Art. 71), the strain in the several pieces compos- 
ing the frame bda may be computed. Again, calculate 
the pressure of that portion of the arch included between a 
and c, and consider half of it collected at b, and acting in a 
vertical direction ; then, by the parallelogram of forces, the 
pressure on the beams bd and bdrnd^y be found. Add the 



JOINTS OF THE ARCII-STONES. 



233 



pressure of that portion of the arch which is included be- 
tween i and b to half the weight of the centre, and consider 
this amount concentrated at d, and acting in a vertical direc- 
tion ; then, by constructing the parallelogram of forces, the 
pressure upon dj may be ascertained. 

The strains having been obtained, the dimensions of the 
several pieces in the frames b a d "^xidi bed may be found by 
computation, as directed in the case of roof-trusses, from 
Arts. 226 to 229. The tie-beams b d, b d, if made of suffi- 
cient size to resist the compressive strain acting upon them 
from the load at b, will be more than large enough to resist 
the tensile strain upon them during the laying of the first 
part of the arch-stones below a and c. 

248.— ArcSi-Stones : Joiiit§. — In an arch, the arch-stones 
are so shaped that the joints between them are perpendicu- 
lar to the curve of the arch, or to its tangent at the point at 
which the joint intersects the curve. In a circular arch, the 




Fig. 1 10. 



joints tend toward the centre of the circle ; in an elliptical 
arch, the joints may be found by the following process: 
To find the direction of the joints for an elliptical arch ; 



/ 


^ 


// 


f 


A/^ 


e \ 


kfY^ 


J " 



Fig. III. 



a joint being wanted at a {Fig. 1 10), draw lines from that 
point to the foci,/ and // bisect the angle /^/ with the 
line ab ; then a b will be the direction of the joint. 



234 CONSTRUCTION. 

To find the direction of the joints for a parabolic arch : 
a joint being wanted at a {Fig. 1 1 1), draw a c ^t right angles 
to the axis eg\ make eg equal to c c, and join a and g\ 
draw ah at right angles to ag\ then ah will be the 
direction of the joint. The direction of the joint from b is 
found in the same manner. The lines ^^and b f are tan- 
gents to the curve at those points respectively ; and any 
number of joints in the curve may be obtained by first 
ascertaining the tangents, and then drawing lines at right 
angles to them. {ScQArt. 462.) 

JOINTS. 

249. — Timljer JJoants. — The joint shown in Fig. 112 is 
simple and strong ; but the strength consists wholly in the 
bolts, and in the friction of the parts produced by screwing 
the pieces firmly together. Should the timber shrink to 



^. 



Fig. 112. 



even a small degree, the strength would depend altogether 
on the bolts. It would be made much stronger by indent- 
ing the pieces together, as at the upper edge of the tie-beam 



C 



-CZ]- 



-■E> Cr- 

FiG. 113. 



in Fig. 113, or by placing keys in the joints, as at the lower 
edge in the same figure. This process, however, weakens 
the beam in proportion to the depth of the indents. 



1 



Fig. 114 

Fig. 1 14 shows a method of scarfing, or splicing, a tie- 
beam without bolts. The keys are to be of well-seasoned, 
hard wood, and, if possible, very cross-grained. The addi- 



THE SPLICING OF TIMBER. 235 

tion of bolts would make this a very strong splice, or even 
white-oak pins would add materially to its strength. 

Fi^. 115 shows about as strong a splice, perhaps, as can 
well be made. It is to be recommended for its simplicity; 
as, on account of there being no oblique joints in it, it can 
be readily and accurately executed. A complicated joint is 
the worst that can be adopted ; still, some have proposed 
joints that seem to have little else besides complication to 
recommend them. 

In proportioning the parts of these scarfs, the depths of 




all the indents taken together should be equal to one third 
of the depth of the beam. In oak, ash or elm, the whole 
length of the scarf should be six times the depth, or thick- 
ness, of the beam, when there are no bolts ; but, if bolts in- 
stead of indents are used, then three times the breadth ; and 
when both methods are combined, twice the depth of the 
beam. The length of the scarf in pine and similar soft 
woods, depending wholly on indents, should be about 12 
times the thickness, or depth, of the beam ; when depend- 
ing wholly on bolts, 6 times the breadth ; and when both 
methods are combined, 4 times the depth. 



Fig. 116. 

Sometimes beams have to be pieced that are required 
to resist cross-strains— such as a girder, or the tie-beam of a 
roof when supporting the ceiling. In such beams, the 
fibres of the wood in the upper part are compressed ; and 
therefore a simple butt joint at that place (as in F{^. 116) 
is far preferable to any other. In such case, an oblique 
joint is the very worst. The under side of the beam being 
in a state of tension, it must be indented or bolted, or both ; 
and an iron plate under the heads of the bolts gives a great 
addition of strengfth. 



236 CONSTRUCTION. 

Scarfing requires accuracy and care, as all the indents 
should bear equally; otherwise, one being strained more 
than another, there would be a tendency to sphnter off the 
parts. Hence the simplest form that will attain the object 
is by far the best. In all beams that are compressed end- 
wise, abutting joints, formed at right angles to the direction 
of their length, are at once the simplest and the best. For a 
temporary purpose, Fig. 112 would do very well ; it would 
be improved, however, by having a piece bolted on all four 
sides. Fig. 113, and indeed each of the others, since they 
have no oblique joints, would resist compression well. 

In framing one beam into another for bearing purposes, 
such as a floor-beam into a trimmer, the best place to make 
the mortise in the trimmer is in the neutral line {Arts. 120, 
121), which is in the middle of its depth. Some have 
thought that, as the fibres of the upper edge are compressed, 



r-j 



Fig. 117. 

a mortise might be made there, and the tenon driven in 
tight enough to make the parts as capable of resisting the 
compression as they would be without it ; and they have 
therefore concluded that plan to be the best. This could 
not be the case, even if the tenon would not shrink ; for a 
joint between two pieces cannot possibly be made to resist 
compression so well as a solid piece without joints. The 
proper place, therefore, for the mortise is at the middle of 
the depth of the beam ; but the best place for the tenon, in 
the floor-beam, is at its bottom edge. For the nearer this 
is placed to the upper edge, the greater is the liability for it 
to splinter off; if the joint is formed, therefore, as at Fig. 117, 
it will combine all the advantages that can be obtained. Dou- 
ble tenons are objectionable, because the piece framed into 
is needlessly weakened, and the tenons are seldom so accu- 
rately made as to bear equally. For this reason, unless the tusk 



THE FRAMING IN A ROOF-TRUSS. 



237 



at a in the figure fits exactly, so as to bear equally with the 
tenon, it had better be omitted. And in sawing the shoulders 
care should be taken not to saw into the tenon in the least, 
as it would wound the beam in the place least able to bear it. 

Thus it will be seen that framing weakens both pieces, 
more or less. It should, therefore, be avoided as much as 
possible , and where it is practicable one piece should rest 
upon the other, rather than be framed into it. This re- 
mark applies to the bearing of floor-beams on a girder, to 
the purlins and jack-rafters of a roof, etc. 

In a framed truss for a roof, bridge, partition, etc., the 
joints should be so constructed as to direct the pressures 
through the axes of the several pieces, and also to avoid 
every tendency of the parts to slide. To attain this object, 





Fig. 118. 



Fig. iig. 



Fig. 120. 



the abuttinsf surface on the end of a strut should be at 
right angles to the direction of the pressure ; as at the joint 
shown in Fig. 118 for the foot of a rafter (see Art. 86), in Fig. 
119 for the head of a rafter, and in Fig. 120 for the foot of a 
strut or brace. The joint at Fig. 118 is not cut completely 
across the tie-beam, but a narrow lip is left standing in the 
middle, and a corresponding indent is made in the rafter, to 
prevent the parts from separating sideways. The abutting 
surface should be made as large as the attainment of other 
necessary objects will admit. The iron strap is added to 
prevent the rafter sliding out, should the end of the tie- 
beam, by decay or otherwise, splinter off. In making the 
joint shown at Fig. 119, it should be left a little open at a, 
so as to bring the parts to a fair bearing at the settling of 
the truss, which must necessarily take place from the shrink- 
ing of the king-post and other parts. If the joint is made 
fair at first, when the truss settles it will cause it to open at 



238 



CONSTRUCTION. 



the under side of the rafter, thus throwing the whole pres- 
sure upon the sharp edge at a. This will cause an indenta- 
tion in the king-post, by which the truss will be made to 
settle further ; and this pressure not being in the axis of the 
rafter, it will be greatly increased, thereby rendering the 
rafter liable to split and break. 



1^-^ 




Fig. 121. 



Fig. 122. 




If the rafters and struts were made to abut end to end, 
as in Figs. I2i, 122 and 123, and the king or queen post 
notched on in halves and bolted, the ill effects of shrinking 
would be avoided. This method has been practised with 
success in some of the most celebrated bridges and roofs in 
Europe ; and, were its use adopted in this country, the un- 
seemly sight of a hogged ridge would seldom be met with. 





Fig. 124. 



Fig. 125. 



A plate of cast-iron between the abutting surfaces will 
equalize the pressure. 

Fig. 124 is a proper joint for a collar-beam in a small 
roof: the principle shown here should characterize all tie- 
joints. The dovetail joint, although extensively practised 
in the above and similar cases, is the very worst that can be 
employed. The shrinking of the timber, if only to a small 



WHITE-OAK PINS AND IRON STRAPS. 239 

degree, permits the tie to withdraw — as is shown at Fig. 
125. The dotted Une shows the position of the tie after it 
has shrunk. 

Locust and white-oak pins are great additions to the 
strength of a joint. In many cases they would supply the 
place of iron bolts ; and, on account of their small cost, they 
should be used in preference wherever the strength of iron 
is not requisite. In small framing, good cut nails are of 
great service at the joints ; but they should not be trusted 
to bear any considerable pressure, as they are apt to be 
brittle. Iron straps are seldom necessary, as all the joinings 
in carpentry may be made without them. They can be 
used to advantage, however, at the foot of suspending-pieces, 
and for the rafter at the end of the tie-beam. In roofs for 
ordinary purposes, the iron straps for suspending-pieces 
may be as follows: When the longest unsupported part of 
the tie-beam is— 

10 feet, the strap may be i inch wide by -f^ thick. 
20 "• " " 2 " " i " 

In fastening a strap, its hold on the suspending-piece will be 
much increased by turning its ends into the wood. Iron 
straps should be protected from rust; for thin plates of iron 
decay very soon, especially when exposed to dampness. 
For this purpose, as soon as the strap is made let it be 
heated to about a blue heat, and, while it is hot, pour over 
its entire surface raw linseed oil, or rub it with beeswax. 
Either of these will give it a coating which dampness will 
not penetrate. 



SECTION III.— STAIRS. 

250. — Stairs : General I&equlrement§. — The STAIRS is 
that commodious arrangement of steps in a building by 
which access is obtained from one story to another. Their 
position, form, and finish, when determined with discrimi- 
nating taste, add greatly to the comfort and elegance of a 
structure. As regards their position, the first object should 
be to have them near the middle of the building, in order 
that they may afford an equally easy access .to all the rooms 
and passages. Next in importance is light ; to obtain which 
they would seem to be best situated near an outer wall, in 
which windows might be constructed for the purpose ; yet 
a skylight, or opening in the roof, would not only provide 
light, and so secure a central position for the stairs, but may 
be made, also, to assist materially as an ornament to the 
building, and, what is of more importance, afford an oppor- 
tunity for better ventilation. 

All stairs, especially those of the most important build- 
ings, should be erected of stone or some equally durable and 
fire-resisting material, that the means of egress from a burn- 
ing building may not be too rapidly destroyed. 

Winding stairs, or those in which the direction is gradu- 
ally changed by means of winders, or steps which taper in 
width, are interesting by reason of the greater skill required 
in their construction ; but are objectionable, for the reason 
that children are exposed to accident by their liability to fall 
when passing over the narrow ends of the steps. Stairs of 
this kind should be tolerated only where there is not suffi- 
cient space for those with flyers, or steps of parallel width. 

Stairs in one long continuous flight are also objection- 
able. Platforms or landings should be introduced at inter- 
vals, so that any one flight may not contain more than about 
twelve or fifteen steps. 

The width of stairs should be in accordance with the im- 




KHORSABAD.-ASSYRIAN TEMPLE, RESTORED.^ ..^ 



THE GRADE OF STAIRS. 241 

portance of the building in which they are placed, varying 
from 3 to 12 feet. Where two persons are expected to pass 
each other conveniently the least width admissible is 3 feet. 
Still, in crowded cities, where land is valuable, the space 
allowed for passages is correspondingly small, and in these 
stairs are sometimes made as narrow as 2^ feet. 

From 3 to 4 feet is a suitable width for a good dwelling ; 
while 5 feet will be found ample for stairs in buildings occu- 
pied by many people ; and from 8 to 12 feet is sufficient for 
the width of stairs in halls of assembly. 

To avoid tripping or stumbling, care should be exer- 
cised, in the planning of a stairs, to secure an even grade. 
To this end, the iioshig, or outer edge, of each step should be 
exactly in line with all the other nosings. In stairs com- 
posed of both flyers and winders, precaution in this regard 
is especially needed. In such stairs, the steps — fiyers and 
winders alike — should be of one width on the line along 
which a person would naturally walk when having his hand 
upon the rail. This tread-X\\\(t, consequently, would be paral- 
lel with the hand-rail, and is usually taken at a distance of 
from 18 to 20 inches from the centre of it. In the plan of 
the stairs this tread-line should be drawn and divided into 
equal parts, each part being the tread, or width of a flyer 
from the face of one riser to the face of the next. 

251.— The Orade of Stairs — The extra exertion required 
in ascending a staircase over that for walking on level 
ground is due to the weight which a person at each step is 
required to lift ; that is, the weight of his own body. Hence 
the difficulty of ascent will be in proportion to the height of 
each step, or to the rise, as it is termed. To facilitate the 
operation of going up stairs, therefore, the risers should be 
low. The grade of a stairs, or its angle of ascent, depends 
not only upon the height of the riser, but also upon the 
width of the step ; and this has a certain relation to the 
riser ; for the width of a step should be in proportion to the 
smallness of the ano:le of ascent. 

The distance from the top of one riser to the top of the 
next is the distance travelled at each step taken, and this dis- 



242 STAIRS. 

tance should vary as the grade of the stairs ; for a person 
who in cUmbing a ladder, or a nearly vertical stairs, can 
travel only 12 inches, or less, at a step, will be able with 
equal or greater facility to travel at least twice this distance 
on level ground. The distance travelled, therefore, should 
be in proportion inversely to the angle of ascent ; or, the di- 
mensions of riser and step should be reciprocal : a low rise 
should have a wide step, and a high rise a narrow step, 

252. — Fitcli-Koard : Relation of Ri§e to Tread. — Among 
the various devices for determining the relation of the rise 
to the tread, or net width of step, one is to make the sum of 
the two equal to 18 inches. 

For example, for a rise of 6 inches the tread should be 
12, for 7 inches the tread should be 1 1 ; or — 

6 + 12 = 18 8 + 10 = 18 
61 + III = 18 8i + 9i= 18 

7 + II = 18 9 + 9 = 18 
7^+ ioi= 18 9J + 8^= 18 

This rule is simple, but the results in extreme cases are not 
satisfactory. If the ascent of a stairs be gradual and easy, 
the length from the top of one rise to that of another, or the 
hypothenuse of the pitch-board, may be proportionally long ; 
but if the stairs be steep, the length must be shorter. 

There is a French method, introduced by Blondel in his 
Coiirs d' Architecture. It is referred to in G wilt's Encyclo- 
pedia, Art. 2813. 

This method is based upon the assumed distance of 24 
inches as being a convenient step upon level ground, and 
upon 12 inches as the most convenient height tt) rise when 
the ascent is vertical. These are French inches, old system. 
The 24 inches French equals about 25-^^ inches English. 

With these distances as base and perpendicular, a right- 
angled triangle is formed, which is used as a scale upon 
which the proportions of a pitch-board are found. For 
example, let a line be drawn from any point in the hypothe- 
nuse of this triangle to the right angle of the triangle ; then 
this line will equal the length of the pitch-board, along the 



PROPORTIONS OF THE PITCH-BOARD. 243 

rake, for a stairs having a grade equal to the angle formed 
by this line and the base-line of the scale. 

In the absence of the triangular scale, the lengths of the 
pitch-boards, as found by this rule, may be computed by this 
expression— 

^=25-j-V-2/^; (107.) 

in which /^equals the tread, or base of the pitch-board, and 
// the riser, or its perpendicular height. 
For example, let // = 6 ; then — 

This result is greater than would be proper in some cases. 

The length of the hypothenuse of the pitch-board should 
be proportional not only to the angle of ascent {Art. 251), but 
also to the strength and height of the class of people who 
are to use the stairs. Tall and strong persons will take 
longer steps than short and feeble people. The hypothe- 
nuse of the pitch-board should be made in proportion to 
the distance taken at a step on level ground by the persons 
who are to use the stairs. 

If people are divided into two classes, one composed of 
robust workmen and the other of delicate women and in- 
firm men, then there may be two scales formed for the pitch- 
boards of stairs — one to be used for shops and factories, and 
the other for dwellings. The distance on level ground trav- 
elled per step, by men, varies from about 26 to 32 inches, or 
on an average 28 inches. The height to which men are 
accustomed to rise on ladders is from 12 to 16 inches at each 
step, or on the average 14 inches. 

With these dimensions, therefore, of 14 and 28 inches, a 
scale may be formed for pitch-boards for stairs, in buildings 
to be used exclusively by robust workmen. And with 12 
and 24 inches another scale may be formed for pitch-boards 
for stairs, in buildings to be used by women and feeble 
people. These two scales are both shown in F/o-. 126. 
They are made thus : Let C A B he ii right angle. Make A 
B equal to 28 inches, and A C equal to 14; then join B and 



244 



STAIRS. 



C. At right angles to C B, from A, draw A F; then with 
A F for radius describe the arc F G. Then a line, as y^ AT 
or A L, drawn from A at any angle with A B and limited by 
the line G F B v^iW give the length of the hypothenuse of 
the pitch-board, for shop stairs of a grade equal to the angle 
which said line makes with A B. From K, perpendicular to 
A B, draw K N\ then K N will be the proper riser for a 
pitch-board of which ^ A^is the tread. So, likewise, L M 
will be the appropriate riser for the tread A M. The arc F G 
is introduced to limit the rake-line of pitch-boards occur- 
ring between F and C, in order to avoid making them longer 
than the one at F. The scale for the stairs for dwellings is 
made in the same manner ; A D — 2\ inches being the base, 
A E =. \2 inches the rise, and J H D the line limiting the 
rake-lines of pitch-boards. 




To compute the length of risers an,d treads, we have for 
the scale for shops, for those occurring between i^and B — 

r:= 1(28-0: (io8.) 

/ = 28 — 2 r ; (109.) 

and for those between F and G, we have — 

(108, A.) 
(109, A.) 



r= 1/156.8 - /^ 



^= |/i56.8-r^ 

For the scale for dweUings, we have, for those occurring 
between H and D — 

r =: 1(24-/); (108, B.) 

/ = 24 — 2 r; (109, B.) 



STAIRS FOR SHOPS AND FOR DWELLINGS. 

and for those between H and J, we have — 



245 



r^ 1/115.2-/^ 



/= 4/115.2 — r'; 



(108, C.) 
(109, C.) 



where, in each equation, r represents the riser, and t the 
tread, or net step. 

By these formulae, the following tables have been com- 
puted : 

Stairs for Shops. 



Rise. 



Tread. Ratio— Rise to Tread. 



24- 
22- 
2T- 

20- 
19. 
18. 
17-20 
16 -60 

i6- 

15-50 

15- 

14-60 

1420 

14- 

13-60 



to 



33 



12 

7' 
6- 

5- 

4-22 

3-6o 

3-19 
.91 



Rise. 


Tread. 


Ratio— Ris2 to Tread, 


7 -40 


13-20 


I to 


1-78 


7-60 


12-80 




1-68 


7-80 


12-40 




1-59 


8- 


12- 




I -50 


8 -20 


II-6 




1-41 


8-50 


II- 




1-29 


8-80 


10-40 




I -18 


9- 


10- 




I- II 


9-30 


9-40 




I-OI 


9-60 


8-80 




0-92 


10 • 


8- 




o-8o 


10-50 


7- 




0-67 


II- 


6- 




0-55 


11-50 


4-95 




0-43 


12- 


3-58 


I" 


0-30 



Stairs for Dwellings. 



Rise. 


Tread. 


Ratio— Rise to Tread. 


Rise. 


Tread. 


Ratio— Rise to Tread. 


2- 


20- 


I to 10- 


7-40 


9 


20 


I to 1-24 


3 




18 




I " 6- 


7 


50 


9 




I " 1-20 


3 


50 


17 




I " 4-86 


7 


60 


8 


80 


I " 1. 16 


4 




16 




I '• 4- 


7 


70 


8 


60 


I " I-I2 


4 


50 


15 




I " 3-33 


7 


80 


8 


40 


I " 1-08 


5 




14 




I •' 2-80 


7 


90 


8 


20 


I " I -04 


5 


40 


13 


20 


I " 2-44 


8 




8 




I " I- 


5 


70 


12 


60 


I " 2-21 


8 


10 


7 


80 


I " 0-96 


6 




12 




I " 2- 


8 


30 


7 


40 


I " 0-89 


6 


25 


II 


50 


I «' 1-84 


8 


50 


7 




I " 0-82 


6 


50 


II 




I " 1-69 


8 


75 


6 


50 


I " 0-74 


6 


75 


10 


50 


I " 1-56 


9 




6 




I " 0-67 


7 




10 




I " 1-43 


9 


30 


5 


40 


I " 0-58 


7 


10 


9 


80 


I '• 1-38 


9 


60 


4 


80 


I " 0-50 


7 


20 


9 


60 


I " 1-33 


10 




3 


90 


I " 0-39 


7-30 


9-40 


I " 1-29 


10-50 


2-20 


I " 0-2I 



These tables will be useful in determining questions in- 



246 STAIRS. 

volving the proportion between the rise and tread of a 
pitch-board. 

For stairs in which the run is limited, to determine the 
number of risers which would give an easy ascent : Divide 
the riui by the Jicight^ and find in the proper table, above, 
the ratio nearest to the quotient^ and in a line with this ratio, 
in the second column to the left, will be found the corre- 
sponding riser. With this divide the rise in inches ; the quo- 
tient^ or the nearest zvhole number thereto, will be the required 
number of risers in the stairs. 

Example. — For the stairs in a dwelling, let the rise be 12' 
8", or 152 inches. Let the run between the extreme risers 
be 17' 2". To this, for the purpose of obtaining the correct 
angle of ascent, by having an equal number of risers and 
treads, add, for one more tread, say 10 inches, its probable 
width; thus making the total run 18 feet, or 216 inches. 
Thus we have for the run 216, and for the rise 152. Divid- 
ing the former by the latter gives i -42 nearly. In the table 
of stairs for dwellings, the ratio nearest to this is i -43, and in 
the line to the left, in the second column, is 7, the approxi- 
mate size of riser appropriate to this case. Dividing the 
rise, 152 inches, by this 7, we have 2 if as the quotient. 

This is nearer to 22 than to 21 ; therefore, the number of 
risers required is 22. 

When the number of risers is determined, then the rise 
divided by this number will give the height of each riser ; 
thus, in the above case, the rise is 152 inches. This divided 
by 22 gives 6-909 inches for the height of the riser. 

When the height of the riser is known, then, if the run is 
unlimited, the width of tread will be found in the proper table 
above. For example, if the riser is 7 inches or nearly that, 
then in the table of stairs for dwellings, in the next column 
to the right, and opposite 7 in the column of risers, is found 
10, the approximate width of tread. By the use of equation 
(109, B.), the width may be had exactly according to the 
scale. For example, equation (109, B.) with 6-91 for the 
riser, becomes— 

/ — 24 — 2 X 6-91 = 10- 18, 
or about io^\ inches. 



TO CONSTRUCT THE PITCH-BOARD. 247 

When the run is limited and the number of risers is 
known, then the width of tread is obtained by dividing the 
run by the number of treads. There are always of treads 
one less than there are of risers, in each flight. 



253. — i>imensions of the Pitch-Board. — The first thing 
in commencing to build a stairs is to make the /zV<://-board ; 
this is done in the following manner: Obtain very accurate- 
ly, in feet and inches, the rise, or perpendicular height, of the 
story in which the stairs are to be placed. This must be 
taken from the top of the lower floor to the top of the upper 
floor. Then, to obtain the number of rises and treads and 
their size, proceed as directed in Art. 252. Having obtained 
these, the pitch-board may be made in the following man- 
ner: Upon a piece of well-seasoned board about |- of an 
inch thick, having one edge jointed 
straight and square, lay the corner 
of a steel square, as shown at Fig. 127. 
Make a b equal to the riser, and b c 
equal to the tread ; mark along the 
edges with a knife, and cut by the ^^'^' ^^7- 

marks, making the edges of the pitch - board perfectly 
square. The grain of the wood should run in the direction 
indicated in the figure, because, in case of shrinkage, the 
rise and the tread will be equally affected by it. When a 
pitch-board is first made, the dimensions of the riser and 
tread should be preserved in figures, in order that, in case 
of shrinkage or damage otherwise, a second may be 
made. 

254. — The String of a Stairs. — The space required for 
timber and plastering under the steps is about 5 inches for 
ordinary stairs, or 6 inches if furred ; set a gauge, there- 
fore, at 5 or 6 inches, as the case requires, and run it on the 
lower edge of the plank, Tis ab {Fig. 128). Commencing at 
one end, lay the longest side of the pitch-board against the 
gauge-mark, a b, as at c, and draw by the edges the lines for 
the first rise and tread ; then place it successively as at d, e, 




248 



STAIRS. 



and y, until the required number of risers shall be laid down. 
To insure accuracy, it is well to ascertain the theoretical 
raking length of the pitch-board by computation, as in note 
to Art. 536, by getting the square root of the sum of the 
squares of the rise and run, and using this by which to 
divide the line ab into equal parts. 




Fig. 128, 

255. — Step and Ri§er Connection. — Fig. 129 represents 
a section of step and riser, joined after the most approved 
method. In this, a represents the end of a block about 2 



UJ. 



Fig, 129. 

inches long, two or three of which, in the length of the 
step, are glued in the corner. The cove at b is planed up 
square, glued in, and stuck or moulded after the glue is 
set. 

PLATFORM STAIRS. 



256. — Platform Stair§ : the Cylinder. — A platform stairs 
ascends from one story to another in two or more flights, 
having platforms or landings between for resting and to 
change their direction. This kind of stairs, being simple, is 



CYLINDER OF PLATFORM STAIRS. 



249 



easily constructed, and at the same time is to be preferred 

to those with ivinders, for the convenience it affords in use 

{Art. 250). The cylinder may be of 

any diameter desirable, from a few 

inches to 3 or more feet, but it is 

generally small, about 6 inches. It 

may be worked out of one solid 

piece, but a better way is to glue 

together 3 pieces, as in Fig. 130 ; in 

which the pieces a, b, and c compose 

the cylinder, and d and e represent 

parts of the strings. The strings, " ^^^' ^'^^' 

after being glued to the cylinder, are secured with screws. 

The joining at and is the most proper for that kind of 

joint. 





Fig. 131, 

257. — Form of I^ower Edge of Cylinder. — Find the 
stretch-out, de {Fig. 131), of the face of the cylinder, a be, 



250 STAIRS. 

according to Art. 524; from d and e draw <^/ and eg at 
right angles to de\ draw kg parallel to de, and make kf 
and ^? each equal to one riser; from i and / draw ij and 
fk parallel to kg] place the tread of the pitch-board at these 
last lines, and draw by the lower edge the lines kh and il\ 
parallel to these draw m n and op, at the requisite distance 
for the dimensions of the string ; from s, the centre of the 
plan, draw sq parallel to df-, divide h q and q g each into two 
equal parts, as at v and w ; from v and w draw v n and w 
parallel to f d\ join n and 0, cutting qs\x\r\ then the angles 
7t7ir and r^'^', being eased off according to Art. 521, will give 
the proper curve for the bottom edge of the cylinder. A 
centre may be found upon which to describe these curves, 
thus : from ti draw u x at right angles to m n ; from r draw 
rx at right angles to 7to\ then x will be the centre for the 
curve ti r. The centre for the curve r t may be found in a 
similar manner. Centres from which to strike these curves 
are usually quite unnecessary ; an experienced workman 
will readily form the curves guided alone by his practised 
eye. 



&. 




Fig. 132. 

258. — Position of the Balusters. — Place the centre of 
the first baluster, b {Fig. 132), half its diameter from the face 
of the riser, cd, and one third its diameter from the end of 
the step, €d\ and place the centre of the other baluster, a, 
half the tread from the centre of the first. A line through 
the centre of the rail will occur vertically over the centres 
of the balusters. The usual length of the balusters is 2 feet 
5 inches and 2 feet 9 inches respectively, for the short and 
long balusters. Their length may be greater than is here 
indicated, but, for safety, should never be less. The differ- 
ence in length between the short and long balusters is 
equal to one half the height of a riser. 



CONSTRUCTION OF WINDING STAIRS. 



251 



259. — ^Viiiding Stairs: have the steps narrower atone 
end than at the other. In some stairs there are steps of 
parallel width incorporated with the tapering steps ; in this 
case the former are called ^j^ers, and the latter winders. 

260. — Itcgiilar Windings Stair§. — In Fig. 133, abed rep- 
resents the inner surface of the wall enclosing- the space 
allotted to the stairs, a e the length of the steps, and e f g h 
the cylinder, or face of the front-string. The line ^ ^ is given 
as the face of the first riser, and the point / for the limit of 




Fig. 133. 



the last. Make e z^ equal to 18 inches, and upon ^, with i 
for radius, describe the arc i j \ obtain the number of risers 
and of treads required to ascend to the floor at j\ according 
to Art. 252, and divide the. arc ij into the same number of 
equal parts as there are to be treads ; through the points of 
division, i, 2, 3, etc., and from the wall-string to the front- 
string, draw lines tending to the centre, o ; then these lines 
will represent the face of each riser, and determine the form 
and width of the steps. Allow the necessary projection for 
the nosing beyond a e, which should be equal to the thick- 



252 STAIRS. 

ness of the step, and then a e I k will be the dimensions for 
each step. Make a pitch-board for the wall-string having a k 
for the tread, and the rise as previously ascertained ; with 
this lay out on a thicknessed plank the several risers and 
treads, as at Fig. 128, gauging from the upper edge of the 
string for the line at which to set the pitch-board. 

Upon the back of the string, with a i^-inch dado plane, 
make a succession of grooves \\ inches apart, and parallel 
with the lines for the risers on the face. These grooves 
must be cut along the whole length of the plank, and deep 
enough to admit of the plank's bending around the curve 
abed. Then construct a drum, or cylinder, of any com- 
mon kind of stuff, made to fit a curve with a radius the 
thickness of the string less than oa ; upon this the string must 
be bent, and the grooves filled with strips of wood, called 
keys^ which must be very nicely fitted and glued in. After 
it has dried, a board thin enough to bend around on the out- 
side of the string must be glued on from one end to the 
other, and nailed with clout-nails. In doing this, be careful 
not to nail into any place opposite to where a riser or step is 
to enter on the face. 

After the string has been on the drum a sufficient time 
for the glue to set, take it off, and cut the mortices for the 
steps and risers on the face at the lines previously made ; 
which may be done by boring with a centre-bit half through 
the string, and nicely chiselling to the line. The drum need 
not be made to extend over the whole space occupied by the 
stairs, but merely so far as requisite to receive one piece of 
the wall-string at a time ; for it is evident that more than 
one Avill be required. The front-string may be constructed 
m the same manner ; taking e I instead of a k for the tread of 
the pitch-board, dadoing it with a smaller dado plane, and 
bending it on a drum of the proper size. 

261. ^Winding §tair§ : Shape and Po§Uioii of Tiiiiber§. — 

The dotted lines in Fig. 133 show the position of the timbers 
as regards the plan ; the shape of each is obtained as follows: 
In Fig. 134, the line i a is equal to a riser, less the thickness 
of the floor, and the lines 2 7^, 3 n, a.o,'^ p, and 6 q are each 



TIMBERS FOR WINDING STAIRS. 253 

equal to one riser. The line a 2 is equal to a in in Fig. 133, 
the line m 3 to m n in that figure, etc. In drawing this 
fiofure, commence at a, and make the lines a i and « 2 of the 
length above specified, and draw them at right angles to 
each other ; draw 2 in at right angles to a 2, and ;;/ 3 at 
right angles to in 2, and make 2 in and in 3 of the lengths 
as above specified ; and so proceed to the end. Then 
through the points i, 2, 3, 4, 5, and 6 trace the line i b ; upon 
the points i, 2, 3, 4, etc., with the size of the timber for 
radius, describe arcs as shown in the figure, and by these 
the lower line may be traced parallel to the upper. This 
will give the proper shape for the timber, a b, in Fig. 133 ; 
and that of the others may be found in a similar manner. In 
ordinary cases, the shape of one face of the timber will be 
sufficient, for a good workman can easily hew it to its 
proper level by that ; but where great accuracy is desirable, 
a pattern for the other side may be found in the same man- 




FiG. 134. 

ner as for the first. In many cases, the timbers beneath cir- 
cular stairs are put up after the stairs are erected, and with- 
out previously giving them the required form ; the work- 
man in shaping them being guided by the form marked out 
by the lower edge of the risers. 

262. — IVinding Stairs urith Flyers : Grade of Front- 
Strini;. — In stairs of this kind, if the winders are confined to 
the quarter circle, the transition from the winders to the 
flyers is too abrupt for convenience, as well as in appear- 
ance. To remove this unsightly bend in the rail and string, 
it is usual to take in among the winders one or more of the 
flyers, and thus graduate the width of the winders to that of 
the flyers. But this is not always done so as to secure the 
best results. By the method now to be shown, both rail and 
strings will be gracefully graded. In Fig. 135, a b repre- 
sents the line of the facia along the floor of the upper story, 



254 STAIRS. 

bee the face of the cylinder, and c d the face of the front- 
string. Make ^<^ equal to \ of the diameter of the baluster, 
and parallel to a b, b e c, and c d draw the centre-line of the 
rail, fg, g h i, and ij\ make gk and ^/ each equal to half the 
width of the rail, and through k and /, parallel to the centre- 
line, draw lines for the convex and the concave sides of the 
rail ; tangical to the convex side of the rail, and parallel to 
k in, draw n o ; obtain the stretch-out, q r, of the semicircle, 
k p lUy according to Art. 524 ; extend a b to t, and k m to s ; 
make c s equal to the length of the steps, and i u equal to 18 
inches, and parallel to m p describe the arcs s t and u 6 ; 
from t draw t w, tending to the centre of the cylinder ; from 
6, and on the line 6 u x, run off the regular tread, as at 5, 4, 
3, 2, I, and v; make 71 x equal to half the arc u 6, and make 
the point of division nearest to x, as v, the limit of the par- 
allel steps, or flyers ; make r equal to m z ; from draw 
«^* at right angles to n 0, and equal to one riser; from a'^ 
draw a'^ s parallel to 710, and equal to one tread; from s, 
through 0, draw s b^. 

Then from zv draw w ^^ at right angles to 71 0, and set up 
on the line w c^^ the same number of risers that the floor. A, 
is above the first winder, B, as at i, 2, 3, 4, 5, and 6 ; through 
5 (on the arc 6 u) draw d'^ e^, tending to the centre of the 
cylinder; from e" draw e^" f- at right angles to no, and 
through 5 (on the line w <:") draw g"^ f^ parallel to it o\ 
through 6 (on the line w c") and /^ draw the line h'^ b'^ \ 
make 6 <;^ equal to half a riser, and from c^ and 6 draw c^- i'^ 
and 67^ parallel to 71 o] make h^ P equal to h'^ f^] from P 
draw P k^ at right angles to i'^ /i^, and from /^ draw f^k^ 
at right angles to f"JP\ upon k'^, \vith k'^ f^ for radius, de- 
scribe the arc f^P\ make b^ P equal to ^'^/^, and ease off 
the angle at /^^ by the curve /-^ P. In the figure, the curve 
is described from a centre, but as this might be imprac- 
ticable in a full-size plan, the curve may be obtained accord- 

* In the references fl^ 6', etc., a new form is introduced for the first time. 
During the time taken to refer to the figure, the memory of the form of these 
may pass from the mind, while that of the sound alone remains; they may 
then be mistaken for a2, b 2, etc. This can be avoided in reading by giving 
them a sound corresponding to their meaning, which is a second^ b secotid^ etc. 



MOULDS FOR QUARTER-CIRCLE STAIRS. 



255 



ing to Art. 521. Then from i, 2, 3, and 4 (on the line zv c-) 
draw lines parallel to 71 0, meeting the curve in in^, ;/^ 0'^, 
and /^; from these points draw lines at right angles to no, 
and meeting it in ;l^^ ^'^ ^^ and /^; from x^ and r'^ draw 



«3 hz 



hi (fz 




Fig. 135. 



lines tending to it -, and meeting the convex side of the rail 
in y^ and ^^; make ;;/ 7^- equal to r s^, and 7/1 w^ equal to 
rt^; from J/-, -a v^^ and w", through 4, 3, 2, and i,draw lines 
meeting the line of the wall-string in a^, d^, c^, and d^ ; from 



256 STAIRS. 

^^ where the centre-line of the rail crosses the line of the 
floor, draw e^ f'^ at right angles to ;z 0, and from/^ through 
6, draw/^-; then the heavy hnes /V^'^^^^ 7^ «^ ^^ ^^ 
v'^c^, w"- d^, and z y will be the lines for the risers, which, 
being extended to the line of the front-string, b e c d, will 
give the dimensions of the winders and the grading of the 
front-string, as was required. 

HAND-RAILING. 

263. — Hand-Railing for Stairs. — A piece of hand-rail- 
ing intended for the curved part of a stairs, when properly 
shaped, has a twisted form, deviating widely from plane sur- 
faces. If laid upon a table it may easily be rocked to and 
fro, and can be made to coincide with the surface of the 
table in only three points. And yet it is usual to cut such 
twisted pieces from ordinary parallel-faced plank ; and to 
cut the plank in form according to a face-vaoxxld, previously 
formed from given dimensions obtained from the plan of the 
stairs. The shape of the finished wreath differs so widely 
from the piece when first cut from the plank as to make it 
appear to a novice a matter of exceeding difficulty, if not an 
impossibility, to design a face-mould which shall cover accu- 
rately the form of the completed wreath. But he will find, 
as he progresses in a study of the subject, that it is not only 
a possibility, but that the science has been reduced to such 
a system that all necessary moulds may be obtained with 
great facility. To attain to this proficiency, however, re- 
quires close attention and continued persistent study, yet no 
more than this important science deserves. The young car- 
penter may entertain a less worthy ambition than that of 
desiring to be able to form from planks of black-walnut or 
mahogany those pieces of hand-railing which, when secured 
together with rail-screws, shall, on applying them over the 
stairs for which they are intended, be found to fit their 
places exactly, and to form graceful curves at the cylinders. 
That railing which requires to be placed upon the stairs 
before cutting the joints, or which requires the curves or 
butt-joints to be refitted after leaving the shop, is discredit- 



PRINCIPLES OF HAND-RAILING. 257 

able to the workman who makes it. No true mechanic will 
be content until he shall be proved able to form the curves 
and cut the joints in the shop, and so accurately that no altera- 
tion shall be needed when the railing is brought to its place 
on the stairs. The science of hand-railing requires some 
knowledge of descriptive geometry — that branch of geometry 
which has for its object the solution of problems involving 
three dimensions by means of intersecting planes. The 
method of obtaining the lengths and bevils of hip and valley 
rafters, etc., as in Art. 233, is a practical example of descrip- 
tive geometry. The lines and angles to be developed in 
problems of hand-railing are to be obtained by methods 
dependent upon like principles. 

264. — Hand-Railing^: Definitions; Planc§ and Solids. 

— Preliminary to an exposition of the method for drawing 
the face-moulds of a hand-rail wreath, certain terms used in 
descriptive geometry need to be defined. Among the tools 
used by a carpenter are those well-known implements called 
planes, such as the jack-plane, fore-plane, smoothing-plane, 
etc. These enable the workman to straighten and smooth 
the faces of boards and plank, and to dress them out of 
iviiid, or so that their surfaces shall be true and unwinding. 
The term plane, as used in descriptive geometry, however, 
refers not to the implement aforesaid, but to the unwinding 
surface formed by these implements. A plane in geometry 
is defined to be such a surface that if any two points in it be 
joined by a straight line, this line will be in contact with the 
surface at every point in its length. With like results lines 
may be drawn in all possible directions upon such a sur- 
face. This can be done only upon an unwinding surface ; 
therefore, a plane is an unwinding surface. Planes are 
understood to be unlimited in their extent, and to pass freely 
through other planes encountered. 

The science of stair-building has to do with prisms and 
cylinders, examples of Avhich are shoAvn in Figs. 136, 137, and 
138. A right prism {Figs. 136 and 137) is a solid standing 
upon a horizontal plane, and with faces each of which is a 
plane. Two of these faces — top and bottom — are horizontal 



258 



STAIRS. 



and are equal polygons, having their corresponding sides 
parallel. 

The other faces of the prism are parallelograms, each of 
which is a vertical plane. When the vertical sides of a 
prism are of equal width, and in number increased indefi- 
nitely, the two polygonal faces of the prism do not differ 
essentially from circles, and thence the prism becomes a 
cylinder. Thus a right cylinder may be defined to be a 
prism, with circles for the horizontal faces {Fig. 138). 



Fig, 136. 



Fig. 137. 



Fig. 138. 



265. — Hand - Railings : Preliminary €on$lderation§. — 

If within the well-hole, or stair-opening, of a circular stairs a 
solid cylinder be constructed of such diameter as shall fill the 
well-hole completely, touching the hand-railing at all points, 
and then if the top of this cjdinder be cut off on a line with 
the top of the hand-railing, the upper end of the cylinder 
would present a winding surface. But if, instead of cutting 
the cylinder as suggested, it be cut by several planes, each 
of which shall extend so as to cover only one of the wreaths 
of the railing, and be so inclined as to touch its top in three 
points, then the form of each of these planes, at its intersec- 
tion with the vertical sides of the cylinder, would present 
the shape of the concave edge of the face-mould for that 
particular piece of hand - railing covered by the plane. 
Again, if a hollow cylinder be constructed so as to be in 
contact with the outer edge of the hand-railing throughout 
its length, and this cylinder be also cut by the aforesaid 



FACE-MOULDS FOR HAND-RAILS. 259 

planes, then each of said planes at its intersection with this 
latter cylinder would present the form of the convex edge 
of the said face-mould. A plank of proper thickness may 
now have marked upon it the shape of this face-mould, and 
the piece covered by the face-mould, when cut from the 
plank, will evidently contain a wreath like that over which 
the face-mould was formed, and which, by cutting away the 
surplus material above and below, may be gradually wrought 
into the graceful form of the required wreath. 

By the considerations here presented some general idea 
may be had of the method pursued, by which the form of a 
face-mould for hand-railing is obtained. A little reflection 
upon what has been advanced will show that the problem 
to be solved is to pass a plane obliquely through a cylinder 
at certain given points, and find its shape at its intersection 
with the vertical surface of the cylinder. Peter Nicholson 
was the first to show how this might be done, and for the 
invention was rewarded, by a scientific society of London, 
with a gold medal. Other writers have suggested some 
sHght improvements on Nicholson's methods. The method 
to which preference is now given, for its simplicity ot work- 
ing and certainty of results, is that which deals with the 
tangents to the curves, instead of with the curves themselves; 
so we do not pass a plane through a cylinder, but through a 
prism the vertical sides of which are tangent to the cylinder, 
and contain the controlling tangents of the face-moulds. The 
task, therefore, is confined principally to finding the tangents 
upon the face-mould. This accomplished, the rest is easy, as 
will be seen. 

The method by which is found the form of the top of a 
prism cut by an oblique plane will now be shown. 

266. — A Prism Cwt toy an ©toHqiic Plane. — A prism is 
shown in perspective at Fig. 139, cut by an oblique plane. 
The points abed are the angles of the horizontal base, and 
abg, bef, edef, and adeg are the vertical sides; while 
^f^S is the top, the form of which is to be shown. 

267. — Form of Top of Prism — In Fig. 139 the form of 
the top of the prism is shown as it appears in perspective, 



26o 



STAIRS. 



not in its real shape ; this is now to be developed. In Fig. 
140, let the sauare « ^ <; <^ represent by scale the actual form 




Fig. 139. 

and size of the base, abed, of the prism shown in Fig, 139. 
Make c, and dd^ respectively equal to the actual heights at 

.4 




Fig. 140. 

cf and de^ Fig. 139 ; the lines dd^ and ee^ being set up per- 
pendicular to the line de. Extend the lines de and d^ e^ until 



ILLUSTRATION BY PLANES. 26 1 

they meet in Jl ; join b and //. Now this line b h is the inter- 
section of two planes : one, the base, or horizontal plane upon 
which the prism stands ; the other, the cutting plane, or the 
plane which, passing obliquely through the prism, cuts it so 
as to produce, by intersecting the vertical sides of the prism, 
the form b feg, Fig. 139. 

To show that b h is the line of intersection of these two 
planes, let the paper on which the triangle dhd^ is drawn 
(designated by the letter E) be lifted by the point d^ and 
revolved on the line dh until d^ stands vertically over d, and 
c^ over c\ then B will be a plane standing on the line dh, 
vertical to the base-plane A . The point h being in the line 
cd extended, and the line ^<^ being in the base-plane ^, there- 
fore h is in the base-plane A. Now the line d^c^ represents 
the line ef oi Fig, 139, and is therefore in the cutting plane ; 
consequently the point //, being also in the line d^c^ ex- 
tended, is also in the cutting plane. By reference to Fig. 
139 it will be seen that the point b is in both the cutting and 
base planes ; we must therefore conclude that, since the two 
points b and h are in both the cutting and base planes, a line 
joining these two points must be the intersection of these two 
planes. The determination of the line of intersection of the 
base and cutting planes is very important, as it is a control- 
ling line ; as will be seen in defining the lines upon which 
the form of the face-mould depends. Care should therefore 
be taken that the method of obtaining it be clearly under- 
stood. 

It will be observed that the intersecting line bh, being in 
the horizontal plane A, is therefore a horizontal line. Also, 
that this horizontal line b h being a line in the cutting plane, 
therefore all lines upon the cutting plane which are drawn 
parallel to b h must also be horizontal lines. The import- 
ance of this will shortly be seen. Through a, perpendicular 
to b Jl, draw the line b^, d^, and parallel with this line draw 
ddj^jj] on d as centre describe the arc d^d^^^/, draw^/^^^^^v 
parallel with dd^^, and extend the latter to d^^, ; on d^, as 
centre describe the arc d^ d^^^ ; join b,, and d^^^. We now 
have three vertical planes which are to be brought into 
position around the base-plane A, as follows: Revolve B 



262 STAIRS. 

upon dh^ E upon dd^^, and C upon b^^ d^^, each until it stands 
perpendicular to the plane A. Then the points d^ and d^^^^ 
will coincide and be vertically over d\ the points d^^^ and d^ 
will coincide and stand vertically over d^^ ; and c^ will cover c. 
These vertical planes will enclose a wedge-shaped figure, 
lying with one face, b^^d^^dh, horizontal and coincident with 
the base-plane A, and three vertical faces, b^^ d^^ d^^^, dd^^ d^ d^^^^, 
and hdd^. By drawing the figure upon a piece of stout 
paper, cutting it out at the outer edges, making creases in 
the lines hd, dd^^, ^u^/n then folding the three planes B, E, 
and C at right angles to A, the relation of the lines will be 
readily seen. Now, to obtain the form of the top or cover to 
the wedge-shaped figure, perpendicular to b^^ d^^^ draw b^^ h^ 
and d^^^e\ on b^^ as centre describe the arc hJi^ ; make d^^^e 
equal to d^^d; join e and h^. Now the form of the top of the 
wedge-shaped figure is shown within the bounds d^^^ b^^ h^ c. 
By revolving this plane D on the line b^^ d^.^ until it is at 
a right angle to the plane C^ and this while the latter is 
supposed to be vertical to the plane A, it will be perceived 
that this movement will place the plane D on top of the 
wedge-shaped figure, and in such a manner as that the point 
e Avill coincide with d^^^^ d^, and the point h^ will fall upon and 
be coincident with the point h, and the lines of the cover 
will coincide with the corresponding lines of the top edges 
of the sides of the figure; for example, the fine b^^d^^/is 
common to the top and the side C] the line d^^^e equals d^^ dy 
which equals d^ d^^^^ ; therefore, the line d^^^ e will coincide 
with ^v<^//// of the side E\ the line eh^ will coincide with dji 
of the side B ; and the line b^^ h, will coincide with the line 
b^Ji. Thus the figure D bounded by b^^d^^^eh^ will exactly 
fit as a cover to the wedge-shaped figure. Upon this cover 
Ave may now develop the form of the top of the prism. 

PreUminary thereto, however, it will be observed, as was 
before remarked, that lines upon the cutting plane which 
are parallel to the intersecting line b^^ h^ are horizontal ; 
and each, therefore, must be of the same length as the line 
in the base-plane A vertically beneath it. For example, the 
line d^^^ e^ is a line in the cutting plane D, parallel with the 
line b^^ h^ in the same plane, and this line b^^ h^ will (when the 



EXPLANATION OF THE DIAGRAMS. 263 

cutting plane D is revolved into its proper position) be co- 
incident with the intersecting Hne b^^ h ; therefore, the line 
d^^^e is a Une in the cutting plane D, drawn parallel with the 
intersecting line b^^ Ji. Now this line d^^^ c, when in position, 
will be coincident with the line d^^^^d^., which lies vertically 
over the line d^^ d of the base-plane A ; its length, therefore, 
is equal to that of the latter. In like manner it may be 
shown that the length of any line on the plane D parallel 
to b^^ hiy is equal in length to the corresponding line upon 
the plane A vertically beneath it. 

Therefore, to obtain the form of the top of the prism, we 
proceed as follows : Perpendicular to b^^ d^ draw c c^^^ and 
aa,ij\ perpendicular to b^^d^^^ draw ^^^^/ and equal to<:^^<:; 
on b^j as centre describe the arc b b^ ; join b^ a^.^, b^f, and a^^^ e. 
Now we have here in plane D the form of the top of the 
prism, as shown in the figure bounded by the lines a^j^b,fc. 
This will be readily seen when the plane D is revolved into 
position. Then the point a^^^ will be vertically over a ; the 
point e coincident with d^ d^^^^ and vertically over <^; the point 
/ coincident with c^ and vertically over c ; while b^ will coin- 
cide with b of the base-plane A. 

The figure a^^^ b^fe, therefore, represents correctly both 
in form and size the top of the prism as it is shown in per- 
spective at bfeg. Fig. 139. The line ef, Fig. 140, is equal to 
the line d^ c^, and so of the other lines bounding the edges of 
the figure. 

The cutting plane bfeg, Fig. 139, may be taken to repre- 
sent the surface of the plank from which the wreath of hand- 
railing is to be cut ; the w^reath curving around from b to e^ 
as shown in Fig. 141, the lines b g and g e being tangent to 
the curve in the cutting plane ; while a b and a d are tan- 
gents to the curve on the base plane, or plane of the cylin- 
der. The location of the cutting plane, however, is usually 
not at the upper surface of the plank, but midway between 
the upper and under surfaces. The tangents in the plane 
are found to be more conveniently located here for deter- 
mining the position of the butt-joints. For a moulded rail 
two curved lines, each with a pair of tangents, are required 
upon the cutting plane, one for the outer edge of the rail, 



264 



STAIRS. 



and the other for the inner edge ; but for a round rail only 
one curve with its tangents is required, as that from ^ to ^ 
in Fig. 141, which is taken to represent the curved line run- 
ning through the centre of the cross-section of the rail. As 
an easy application of the principles regarding the prism, 
just developed, an example will now be given. 




268. — Face-Hould for Hand-Railings of Platform Stairs. 

— Lety /^ and / vi, Fig. 142, represent the central or axial lines 
of the hand-rails of the two flights, one above, the other be- 
low the platform ; and let the semicircleyV/ be the central 
line of the rail around the cylinder at the platform, the risers 
at the platform being located at/ and /. Vertically over the 
platform risers draw ^^^ ; make gr^ equal to a riser of the 
lower flight, and r^g^ and ss^ each equal to a riser of the 
upper flight. Draw g^s and gk^ horizontal and equal 
each to a tread of each flight respectively. Through r^ draw 
k, a^^y and through g^ draw Si t^. Vertically over d draw a^ t^. 
Horizontally draw a^^ a^^^, and t^ t^^. 

It is usual to extend the wreath of the cylinder so as to 
include a part of the straight rail — such a part as convenience 
may require. Let the straight part here to be included ex- 
tend from / to ^ on the plan. Vertically over b draw b^ c^^^y 
and horizontally draw b^ w^^ ; at any point on b^ w^^ locate w^^, 
and make w^^w^ equal to //, and bisect it in iv ; erect the 
perpendiculars w^ a^^^y w d^^^, and w^^ v ; join t^^ and a^^^^ ; from 
^v// horizontally draw d^^^ d^^ ; parallel with r^ k^ draw 
^v/ Cii.r W^ ^^^ have the plan and elevations of the prism. 



FACE-MOULD FOR PLATFORM STAIRS. 



265 



containing at its angles the tangents required for the wreath 
extending from ^o ^ on the plan. The elevation i^ is a view 
of the cylinder looking in the direction dc. 




Fig. 142. 

Comparing Fig. 142 with Fig. 141, the line h^zv^, is the 
trace, upon a vertical plane, of the horizontal plane abed 



266 STAIRS. 

of Fig. 141, or is the ground-line from which the heights of 
the prism are to be taken. 

The triangle <^;, b^ a^^ is represented in Fig. 141 at ab g, and 
the inclined hne b^ a^^ is the tangent of the rail of the lower 
flight, and is represented in Fig. 141 at b g ; while a^^^^ t^^ is 
the tangent of the railing around the cylinder, and the half 
of it is represented in Fig. 141 at ge. The height b^c^^^^ is 
shown in Fig. 141 at cf, while the height w d^^^, or a^ d^^, is 
shown in Fig. 141 at <^^. 

The vertical planes B F C may now be constructed about 
the prism as in />^. 140, proceeding thus: Make cc^ equal to 
b^ Cjjji, and dd^ equal to a^ d^^ ; through c^ draw d^ h ; through 
b draw // b^^ ; perpendicular to // b^^ through a draw b^^ dy ; 
from <^ parallel with /^^^ </vdraw dd^^^^ ; on <:/ as centre describe 
the arc d^d^^^/, draw <^^^^^ d^, also d d^^^, parallel with hb^/, 
on d^j as centre describe the arc dyd^^^ ; join d^^^ to b^^. Par- 
allel with bj^ h draw from each important point of the plan, 
as shown, an ordinate extending to the line b^^ d^^^, and thence 
across plane D draw ordinates perpendicular to b^^ d^^^, and 
make them respectively equal to the corresponding ordinates 
of the plane A, measured from the line b^^ d^ ; join e to f, a^^^ 
to b^, a^^^ to e, and b^ to/; also join /^ to r ^. Then a^^^ b^ is the 
tangent standing over a b, and a^.^ e is the tangent standing 
over ad. The line bj^ is the part of the tangent which 
stands over b l^, the portion of the wreath which is straight. 
The curve en^p^l^ is the trace upon the cutting plane of the 
quarter circle d7ipl, traced through the points 71 ^p^, and as 
many more as desirable, found by ordinates as any other 
point in the plane A. Thus we have complete the line 
bj, n^e, the central hne of the wreath extending from b to d 
in the plan. This is the essential part of the face-mould, which 
is now to be drawn as follows: At Fig. 143 repeat the par- 
allelogram a^^. bjfe of Fig. 142, and, with a radius equal to 
half the diameter of the rail, describe, from centres taken on 
the central line, the several circles shown ; and tangent to 
these circles draw the outer and inner edges of the rail. 
The joint at b^ is to be drawn perpendicular to the tangent 
b^a^^^, while that at e is to be perpendicular to the tangent 
e a^^^. This completes the face-mould for the wreath over 



WREATHS FOR A ROUND RAIL. 267 

blndoi the plan. If the pitch-board of the upper flight be 
the same as that of the lower flight, the face-mould at Fig. 
143 will, reversed, serve also for the wreath over the other 
half of the cylinder. 

In using this face-mould, place it upon a plank equal in 
thickness to the diameter of the rail, mark its form upon the 
plank, and saw square through ; then chamfer the wreath to 
an octagonal form, after which carefully remove the angles 
so as to produce the required round form. The joints, as well 
as the curved edges, are to be cut square through the plank. 

Many more lines have been used in obtaining this face- 
mould than were really necessary for so simple a case, but no 
more than was deemed advisable in order properly to eluci- 
date the general principles involved. A very simple method 

an 




Fig. 143. 



for fage-moulds of platform stairs with small cylinders will 
now be shown. 

269. — More iSimpIe Method for Mand-Rafl to Platform 
Stairs. — In Fig. 144, j'ge represents a pitch-board of the first 
flight, and d and i the pitch-board of the second flight of a plat- 
form stairs, the line <?/ being the top of the platform ; and 
adc is the plan of a line passing through the centre of 
the rail around the cylinder. Through i and d draw i i% 
and through y and e draw 7 /C'; from k draw k I parallel to fc ; 
from b draw bin parallel to gd ; from / draw Ir parallel to 
kj; from n draw ^^ / at right angles to j k ; on the line ob 
make ot equal to 7tt ; join c and t ; on the line jc, Fig. 145, 
make ec equal to en at Fig. 144 ; from c draw ^ ^^ at right 
angles to j c, and make ct equal to ^ / at Fig. 144; through / 
draw p I parallel to j c, and make 1 1 equal to / / at Fig. 144 ; 
join /and c, and complete the parallelogram e cls\ find the 
points 0, 0, 0, according to Art. 551 ; upon e, 0, 0, 0, and /, 



268 



STAIRS. 



successively, with a radius equal to half the width of the 
rail, describe the circles shown in the figure ; then a curve 
traced on both sides of these circles, and just touching them, 




Fig. 144. 



will give the proper form for the mould. The joint at / is 
drawn at right angles to c I, 




Fig. 145. 

This simple method for obtaining the face-moulds for the 
hand-rail of a platform stairs appeared first in the early edi- 
tions of this work. It was invented by a Mr. Kells, an 



HAND-RAIL TO PLATFORM STAIRS. 



269 



eminent stair-builder of this city. A comparison with Fig, 
142 will explain the use of the few lines introduced. For a 
full comprehension of it reference is made to Fig. 146, in 
which the cylinder, for this purpose, is made rectangular 




Fig. 146. 

instead of circular. The figure gives a perspective view of 
a part of the upper and of the lower flights, and a part of 
the platform about the cylinder. The heavy lines, im, mc, 
and cj, show the direction of the rail, and are supposed to 
pass through the centre of it. Assuming that the rake of 



270 



STAIRS. 



the second flight is the same as that of the first, as is gener- 
ally the case, the face-mould for the lower twist will, when 
reversed, do for the upper flight ; that part of the rail, there- 
fore, which passes from c to <:, and from c to /, is all that will 
need explanation. 

Suppose, then, that the parallelogram eaoc represent a 
plane lying perpendicularly over eabf, being inclined in 
the direction e c^ and level in the direction co ; suppose this 




Fig. 147. 

plane eaoch^ revolved on ^^ as an axis, in the manner indi- 
cated by the arcs on and ax, until it coincides with the 
plane ertc\ the line ao will then be represented by the fine 
xn ; then add. the parallelogram xrtn, and the triangle ctl, 
deducting the triangle er s\ then the edges of the plane cslc, 
inchned in the direction ec, and also in the direction c I, will 
lie perpendicularly over the plane eabf. From this we 
gather that the line co, being at right angles to ec, must, in 



HAND-RAIL FOR LARGE CYLINDER. 



271 



order to reach the point /, be lengthened the distance nt, 
and the right angle ^r/ be made obtuse by the addition to 
it of the angle tcL By reference to Fig. 144, it will be seen 
that this lengthening is performed by forming the right- 
angled triangle cot^ corresponding to the triangle cot in 
Fig. 146. The line c t 'vs> then transferred to Fig. 145, and 
placed at right angles to ^<f ; this angle e ct\s then increased 
by adding the angle tcl, corresponding to tcl, Fig. 146. 
Thus the point / is reached, and the proper position and 
length of the lines ec and <;/ obtained. To obtain the face- 
mould for a rail over a cylindrical well-hole, the same process 
is necessary to be followed until the length and position of 
these lines are found ; then, by forming the parallelogram 
eels, and describing a quarter of an ellipse therein, the 
proper form will be given. 




Fig. 14S. 



270b — MaBid-SSaMing foi* a L.arger Cylinder. — Fig. 147 
represents a plan and a vertical section of a line passing 
throusfh the centre of the rail as before. From b draw bk 
parallel to cd\ extend the lines /^andyV until they meet kb 
in k and /; from ;/ draw nl parallel X.0 ob\ through / draw 
It parallel to y /I' ; f rom y^ draw /^ / at right angles toy /C'; on 
the line ob make ot equal to kt. Make ec {Fig. 148) equal 
to ^/^ at Fig. 147 ; from c draw c t Tst right angles to e c, and 
equal to ct at Fig. 147 ; from t draw // parallel to c c, and 
make tl equal to //at Fig. 147 ; complete the parallelogram 
eels, and find the points o, 0, o, as before ; then describe the 
circles and complete the mould as in Fig. 145. The difference 
between this and Case i is that the line et, instead of beins: 
raised and thrown out, is lowered and drawn in. A method 
of planning a cylinder so as to avoid the necessity of cant- 
ing the plank, either up or down, will now be shown. 



2/2 



STAIRS. 



271. — Face-MouM xvUhout Canting the Plank. — Instead 
of placing the platform-risers at the spring of the cylinder, a 
more easy and graceful appearance may be given to the 
rail, and the necessity of canting either of the twists entirely 
obviated, by fixing the place of the above risers at a certain 
distance within the cylinder, as shown in Fig. 149 — the lines 
indicating the face of the risers cutting the cylinder at k and 
/, instead of at / and q, the spring of the cylinder. To 
ascertain the position of the risers, let abc be the pitch- 
board of the lower flight, and cde that of the upper flight, 

these being placed so that b c and 
cd shall form a right line. Extend 
^ ^ to cut ^^ in y ; draw fg parallel 
to db^ and of indefinite length ; 
draw go at right angles to f g, and 
® equal in length to the radius of the 
circle formed by the centre of the 
rail in passing around the cylinder ; 
on as centre describe the semi- 
circle /^^'y through draw is par- 
allel to <^^; make oh equal to the 
radius of the cylinder, and describe 
on the face of the cylinder phq\ 
then extend db across the cylinder, 
cutting it in / and k — giving the 
position of the face of the risers, 
as required. To find the face- 
mould for the twists is simple and 
obvious : it being merely a quarter 
of an ellipse, having j for semi- 
minoraxis, and sf for the semi-major axis; or, at Fig. 151, 
let dcih^^i right angle ; make c i equal to oj, Fig. 149, and dc 
equal to sf, Fig. 149; then draw do parallel to ci, and com- 
plete the curve as before. 





X 


Hs 






^ 


f 


A 


^c ■■■"■■■ 


e 


1 


A 


I 

1 
1 








j 


i« .^ 




p- 


H 




0- 




S 




i 





Fig. 149. 



272- — Railing for Platform IStairs where the Rake 
meets the L.evel. — In Fig. 150, abc\'s> the plan of a line pass- 
ing through the centre of the rail around the cylinder as 
before, and je is a vertical section of two steps starting 
from the floor 



, hg. 



Bisect eJi in d, and through d draw df 



HAND-RAIL AT RAKE AND LEVEL. 



273 



parallel to hg\ bisect f n in /, and from / draw It parallel 
to nj \ from n draw nt at right angles to jn ; on the line ob 
make ot equal to n t. Then, to obtain a mould for the twist 
going up the flight, proceed as at Fig. 145 ; making ^^ in 
that figure equal to en in Fig. 150, and the other lines of a 
length and position such as is indicated by the letters of 
reference in each figure. To obtain the mould for the level 




Fig. 150. 



rail, extend bo (Fig. 150) to i ; make o i equal to //, and join 
/ and ^ ; make^2(/^^> 151) equal to ci Tit Fig. 150; through 




Fig. 151. 

c draw c d ?it right angles to ci\ make dc equal to df at 
Fig. 150, and complete the parallelogram odci\ then pro- 
ceed as in the previous cases to find the mould. 



273. — Application of Face-Moulds to Plank. — All the 

moulds obtained by the preceding examples have been for 
round rails. For these, the mould may be applied to a plank 
of the same thickness as the rail is intended to be, and the 



2/4 



STAIRS. 



plank sawed square through, the joints being cut square 
from the face of the plank. A twist thus cut and truly 
rounded will hang in a proper position over the plan, and 
present a perfect and graceful wreath. 



274. — Face-Moulds for Moulded Rails upon Platform 
Stairs. — In Fig. 152, ^ ^^ is the plan of a line passing through 




Fig. 152. 

the centre of the rail around the cylinder, as before, and the 
lines above it are a vertical section of steps, risers, and plat- 
form, with the lines for the rail obtained as in Fig. 144. Set 
half the width of the rail from b to f and from b to r, and 
from / and r draw fe and r d parallel to ca. At Fig. 153 
the centre-lines of the rail jc and c I are obtained as in the 
^previous examples, making Jc equal jn of Fig. 152, ct 



KACE-MOCLD APPLIED TO PLANK. 2/5 

rou-^l ct of Fi-. 152, and tl equal si of Fig. 152. Make r ^■ 
Jd ; each eV.al to .. at mg. 152. and draw the lines ... 
and kg parallel to .y ; make /. and /r equal to « c and « rf at 
Fk .£ and draw rf« and eq parallel to /.; also, through j 
driwL- parallel to Ic; then, in the parallelograms «.;/r. 
and ..;., f^nd the elliptic curves, dm and ^S.-^-o.^^U. 
Art\^x, and they will define the curves. The fine dp, 
being drawn through / perpendicular to /r, defines the 
joint which is to be cut square through the plank. 

275.— Application of Face-MouKU to Planli.— In Fig. 
,S- make a drawing, from ./ to //, of the cross-section ol the 
hand-rail, and tangent to the lower corner draw the nie g li 
The distance between the lines 7> and g h is the thickness of 
the plank from which the rail is to be cut. Lay the face- 
mould upon the plank, mark its shape upon the plank, and 




Fig. 153- 

saw it square through. To proceed strictly in accordance 
with the requirements of the principles upon which the lace- 
mould is formed, the cutting ought to be made vertically 
through the plank, the latter being in the position which it 
would occupy when upon the stairs. Formerly it was the 
custom to cut it thus, with its long raking lines. But, owing 
to the great labor and inconvenience of this method, efforts 
were made to secure an easier process. By investigation it 
was found that it was possible, without change in the face- 
mould, to cut the plank square through and still obtain the 
correct figure for the raiUng, and this method is the one now 
usually pursued. Not only is the labor of sawing much re- 
duced by this change ; but to the workman it is an entire re- 
lief, as he now, after marking the form of the wreath upon the 
plank, sends it to a steam saw-mill, and, at a small cost, has it 



!76 



STAIRS. 



cut out with an upright scroll-saw. When thus cut out in 
the square, the upper surface of the plank is to be faced up 
true and unwinding, and the outer edge jointed straight 
and square from the face. Then a figure of the cross-section 
of the hand-railing is to be carefully drawn on the ends of the 
squared block as shown in Figs. 154 and 155, and w^hich 
are regulated so as to be correctly in position, as follows. 
First, as to the end Ji of the straight part hj\ In Fig. 154, 
X^tab c dh^ an end view of the squared block, of which a cfd 
is the shape of the end of the straight part. Let the point g 
be the centre of this end of the straight part ; through g 
draw upon the end a e f d the line j k, so that the angle bjk 
shall be equal to the angle kt c, Fig. 152. This is the angle 
at which the plank is required to be canted, revolving it on 





Fig. 154. 



Fig. 155. 



the axis of the straight part of the rail. Through g draw 
the line n h parallel with a h. Upon a thin sheet of metal 
(zinc is preferable) mark carefully the exact figure of the 
cross-section of the rail, drawing a vertical line through its 
centre, cut away the surplus metal, then, with this template 
as a pattern, mark upon the end a efd, Fig. 154, the figure of 
the rail as show^n, the vertical line upon the template being 
made to coincide with the Wnej'k. From n and h draw the 
vertical lines h in and In parallel with j k. 

Now, as to the other end of the square block : Let b cfe. 
Fig. 155, represent the block, of which bcvn is the form of 
the end at the curved part, and o its centre. Through 
dra^v/^, so that the angle e pq shall be equal to the angle 
jnb, Fig. 152. Also, through draw d h parallel with e b\ 



CUTTING THE TWIST-RAIL. 



277 




from ^and Ji draw the vertical lines Ji rand ds parallel with 
pq. Place the template on bcvn, the end of the block, so 
that the vertical line throusfh its centre shall coincide with 
pq; mark its form, then from j, at mid-thickness, draw iv y 
parallel with / q. 

In applying the mould, let Fig. 156 represent the upper 

face of the squared block, .^ x 

with the face-mould lying ^ 
upon it. With the distance 
a /, Fig. 154, and by the 
edge a x, mark a gauge-line 
upon the upper face of the Fig. 156. 

squared block. Set the outer edge of the lace-mould to coin- 
cide with this gauge-line. Let the end of the face-mould be 
set at zv, e iv being equal to e zv, Fig. 155; then mark the 
block by the edge of the face-mould. 

Now turn the block over and apply the face-mould to the 
underside, as in Fig. 157. With the distance d i?i, Fig. 154, 
and by the outer edge of 
the block, mark a gauge- 
line from m, Fig. 157. Set 
the inner edge of the face- 
mould to this gauge-line, 
and slide it endwise till the 
distance em shall equal ezu, Fig. 155, then mark the block by 
the edges of the face-mould. The over wood may now be re- 
moved as indicated by the vertical lines at the sides of the 
cross-section marked on each end of the block (see also Fig. 
167) : the direction of the cutting at the curves must be verti- 
cal ; the inner curve will require a round-faced plane. A com- 
parison of the several figures referred to, with the directions 
given, together with a little reflection, will manifest the 
reasons for the method here given for applying the face- 
mould. Especially so when it is remembered that the face- 
mould was obtained not for the top of the rail, but for the rail 
at the mid-thickness of the block. So, therefore, in the 
application to the upper surface of the block, the face-mould 
is slid up the rake far enough to put the mould in position 
vertically over its true position at mid-thickness ; and on the 



•m 




Fig. 157. 



2/8 STAIRS. 

contrary, in applying the face-mould to the underside of the 
plank, it is slid down until it is vertically beneath its true 
position at the mid-thickness of the block. 

When the vertical faces are completed,, the over wood 
above and below the wreath is to be removed. In doing 
this, the form at the ends, as given by the template, is a suf- 
ficient guide there. Between these the upper and under 
surfaces are to be warped from one end to the other, so as 
to form a graceful curve. With a little practice an intelli- 
gent mechanic will be able to work these surfaces with 
facility. The form of cross-section produced by this opera- 
tion is that of a parallelogram, tangent to the top, bottom, 
and two sides of the rail ; and which at and near the ends 
of the block is not quite full. The next operation is that of 
working the moulding at the sides and on top, first by re- 
bates at the sides, then chamfering, and finally moulding the 
curves. Templates to fit the rail, one at the sides, another 
on top, are useful as checks against cutting away too much 
of the wood. 

The joints are all to be worked square through the plank 
in the line drawn perpendicular to the tangent, as shown in 
F^g' 153- 

276. — Hand-Railing for Circular Stairs. — Let it be re- 
quired to furnish the face-moulds for a circular stairs similar 
to that shown in Fig. 133. 

Preliminary to making the face-moulds it is requisite to 
make a plan, or horizontal projection of the stairs, and on 
this to locate the projections of the tangents and develop 
their vertical projections. For this purpose let bcdefg, 
Fig. 158, be the horizontal projection of the centre of the 
rail, and the lines numbered from 1 to 19 be the risers. At 
any point, a, on an extension of the line of the first riser 
locate the centre of the newel. On <^ as a centre describe the 
two circles ; the larger one equal in diameter to the diame- 
ter of the newel-cap, the inner one distant from the outer 
one equal to half the width of the rail. Let the first joint in 
the hand-rail be located at b, at the fourth riser ; through b 
draw h k tangent to the circle. Select a point, //, on this 



PLAN OF CIRCULAR STAIRS. 



279 



tangent which shall be equally distant from b and from the 
inner circle of the newel-cap, measured on a line tending to 
a ; join h and a^ and from a point, ^, on the line b describe 




Fig. 158. 

the curve from b to the point of the mitre of the newel-cap, 
the curve being tangent, at this point, to the line a h. Select 
positions for the other joints in the hand-rail as at c, d, c, and /. 



280 STAIRS. 

Through these draw lines tangent to the circle.* Then the 
horizontal projection of the tangents will be the lines Jik, kl, 
I in, 1/1 11, and np. Now, if a vertical plane stand upon each 
of these lines, these planes would form a prism not quite 
complete standing upon the base-plane, A. Upon these ver- 
tical planes, C, D, E, F, G, and H, lines may be drawn which 
at each joint shall be tangent to the central line of the rail. 
These are the tangents now to be sought. Perpendicular to 
the tangents at b, c, d, etc., draw the lines bb^,cc^,dd^,ee^, ff^, 
ggi, and h h^^, k k^, k k^^,l l^, 1 1^,, etc. As b is at the fourth riser, 
and the height is counted from the top of the first riser, 
make b b^ equal to three risers. (To avoid extending the 
drawing to inconvenient dimensions, the heights in it are 
made only half their actual size. As this is done uniformly 
throughout the drawing, this reduction will lead to no error 
in the desired results.) As c is on the eighth riser, therefore 
make c c^ equal to seven risers, and so, in like manner, make 
the heights dd^,ee„ and ff^ each of a height to correspond 
with the number of the riser at which it is placed, deduct- 
ing one riser. These heights fix the location of each tangent 
at its point of contact with the central line of the rail. But 
each tangent is yet free to revolve on this point of contact, 
up or down, as may be required to bring the ends of each 
pair of tangents in contact ; or, to make equal in height the 
edges of each pair of vertical planes, which coincide after 
they are revolved on their base-lines into a vertical position ; 
as, for example : the edges k k^ and k k^j of the planes C and 
D must be of equal height; so, also, the edges //, and //^, of 
the planes D and E must be of equal height. The method 
of establishing these heights will now be shown. 

To this end let it be observed, that of the horizontal pro- 
jection of any pair of intersecting tangents, their lengths, 
from the point of intersection to the points of contact with 
the circle, are equal ; for example : of the two tangents h k 
and Ik, the distances from k, their point of intersection, to b 
and c, their points of contact with the circle, are equal ; and 
so also r/ equals d I, dm equals e in, etc. It will be observed 

* A tangent is a line perpendicular to the radius, drawn from the point of 
contact. 



THE FALLING-MOULD FOR THE RAIL. 28 1 

that this equality is not dependent on b^ c, d, etc., the points 
of contact, being disposed at equal distances ; for, in this 
example, they are placed at unequal distances, some being- at 
three treads apart and others at four ; and yet while this un- 
equal distribution of the points b, c, d, etc., has the effect of 
causing the point of contact, as b, c, or c, to divide each whole 
tangent into two unequal parts, it does not disturb the 
equality of the two adjoining parts of any two adjacent tan- 
gents. Now, because of this equality of the two adjoining 
parts of a pair of tangents, the height to be overcome in 
passing from one point of contact to the next must be 
divided equally between the two ; each tangent takes half 
the distance. Therefore, for stairs of this kind, the arrange- 
ment being symmetrical, we have this rule by which to fix 
the height of the ends of any two adjoining tangents, namely : 
To the height at the lower point of contact add half the dif- 
ference between the heights at the two points of contact ; 
the sum will be the required height of the two adjoining 
ends of tangents. For example: the heights at b and r, 
two adjacent points of contact, are respectively three and 
seven risers ; the difference is four risers ; half this added to 
three, the height of the lower rise, gives five risers as the 
height of k k^, kkjj, the height at the adjoining ends of the 
tangents h k and Ik. Again, the heights at c and d are re- 
spectively seven and ten risers ; their difference is three ; 
half of which, or one and a half risers, added to seven, the 
height at the lower point of contact, makes nine and a half 
risers as the heights //^, //^^, at the ends of the adjoining 
tangents k I and ;;/ /. In a similar manner are established 
the heights of the tangents at in, ;/, and /. 

The rule for finding the heights of tangents as just given 
is applicable to circular stairs in Avhich the treads are di- 
vided equally at the front-string, as in Fig. 158. Stairs of 
irregular plan require to have drawn an elevation of the 
rail, stretched out into a plane, upon which the tangents can 
be located. This will be shown farther on. 

The locations of the joints c, d, c, in this example, were 
disposed at unequal distances merely to show the effect on 
the tangents as before noticed. In practice it is proper to 



282 STAIRS. 

locate them at equal distances, for then one face-mould in 
such a stairs will serve for each wreath. 

When the tangent at G has been drawn, the level tangent 
for the landing maybe obtained in this manner: As the 
joint /is located at the eighteenth riser, one riser below the 
landing, draw a horizontal line at ^, one riser above the point 
/, and at half a riser above this draw the level line at /^ ; then 
this line is the level tangent, and p its point of intersection 
with the raking tangent. Draw the vertical line/^/, and 
from/ draw the tangent/^, w^hich is the horizontal projec- 
tion of the tangent p^ g, on plane H (which, to avoid undue 
enlargement of the drawing, is reduced in height), where 
//// equals// . 

To obtain the horizontal tangent t il at the newel, pro- 
ceed thus : Fix the point r, in the tangent r k^, at a height 
above b t equal to the elevation of the centre of the newel 
above the height of a short baluster — for example, from 5 
to 8 inches- — and draw a line through r parallel to b t ] this 
is a horizontal line through the middle of the height of the 
newel-cap, and upon which and the rake the easement to 
the newel is formed. Perpendicular to b t draAv r t, and join 
/ and 7c ; then t it is the horizontal tangent. 

277.— Face-Moulds for Circular Stair§. — At Fig. 159 the 
plan of the newel and the adjacent hand-rail are repeated, 
but upon an enlarged scale ; and in which b b^ is the reduced 
height of the point by or is equal to bb^ less t r, Fig. 158, 
and the angle bb^ ^equals the angle bb^r of Fig. 158. In 
this plan the actual heights must now be taken. Join / and 
iL ; then 1 7t is the level tangent, as also the line of intersection 
of the cutting plane C and the horizontal plane A. Perpen- 
dicular to 1 71, at a point t or anywhere above it, draw u^ b^^. 
Parallel with 1 71 draw b b^^^ ; make b^^ b^^^ equal to b b/, join 
bjjj and 71^ ; then the angle b b^^^ 71^ is the angle which the 
plank in position makes with a vertical line, or what is 
usually termed the phuitb-bevil. Perpendicular to b^^^ 7i^ 
draw 7L^ 71^^ and b^^^ b^^^^ ; make b^^, b^^^^ equal to b b^^ ; make 71^ 
t^ equal to 7t^ /, and 71^ 71^^ to 71^ 71 ; join b^^^^ and /, ; then b^^^^ t^ is 
the tangent in the cutting plane, the horizontal projection of 



FACE-MOULD FOR FIRST SECTION 



283 



which is b t. The butt-joint at b^.^^ is drawn square to the 
tangent b^^^^ t ^. Parallel to the intersecting line / ti, draw 
ordinates across the plane A from as many points as desir- 
able, and extend them to the rake-line 7i, b^^^ ; through the 
points of their intersection with this line, and perpendicular 
to it, draw corresponding ordinates across the plane C. Make 
^.i^ui equal to<^; d, and so in like manner, for all other points, 




Fig. 159. 

obtain in the plane C for each point in the horizontal plane 
A its corresponding point in the plane C \ in each case 
taking the distance to the point in the plane A from the line 
Jl b^i and applying it in the plane C from the rake-line ti^ b^^^. 
For the curves bend a flexible strip to coincide with the 
several points obtained, and draw the curve by the side of 
the strip. The point of the mitre is at d,,^, the mitre-joint is 



284 STAIRS. 

shown at hd^^^ and d^^^ c^^. The line f c^^ is drawn through 
c^j, the most projecting point of the mitre, and parallel to the 
rake-line tt^ b^^^. Additional wood is left attached, extending 
from h to /; this is an allowance to cover the mitre, which 
has to be cut vertically ; the butt-joint at b^^^^ and the face at 
f Cj, are both to be cut square through the plank. The face 
fCjj^ because it is parallel to the rake-line u^ b^^^, is a vertical 
face, as well as being perpendicular to the surface of the 
plank. On it, therefore, lines drawn according to the rake, 
or like the angle ?/^ b^^^ b^^, will be vertical and will give the 
direction of the mitre-faces. We now have at C the face- 
mould for the railing over the plan from b io d \n A, The 
mould thus found is that made upon a cutting plane C, passed 
through the plank, parallel to its face, but at the middle of 
its thickness. To put it in position, let the plane C be lifted 
by its upper edge c^^ and revolved upon the line u^ b^^^ until 
it stands perpendicular to the plane B. Now revolve both 
C and B (kept in this relative position during the revolution) 
upon the line zti b^^ until the plane B stands perpendicular to 
the plane A. Then every point upon plane C will be verti- 
cally over its corresponding point in the plane A. For ex- 
ample, the point b^^^^ will be vertically over b, t^ over t, 
and so of all other points. To show the application of the 
face-mould to the plank, make b^^^ b^. equal to half the thick- 
ness of the plank; parallel to ti^ b^^^ draw b^Cy a line which 
represents the upper surface of the plank, for the Une it^ b^^, 
is at the middle of the thickness. Through b^^^^, and parallel 
with b^^^ 21., draw the line c^ b^^^^ and extend it across the face- 
mould ; make b^^^^ c^ equal \.ob^c\ through c^, and parallel with 
b^^^^ tj, draw c^ e. Now, m n o^p is an end view of the plank, 
showing the face view of the butt-joint at b^^^^. Through r, 
the centre, draw a line parallel with the sides. Then b^.-^ rep- 
resents the point b^^^^ ; make b^-^ e^ equal to b^^^^ e ; through r, 
the centre, draw c, r across the face of the joint ; then e^ r is 
a vertical line (see Art. 284), parallel and perpendicular to 
which the four sides of the squared-up wreath are to be 
drawn as shown. In applying the face-mould to the plank at 
first, for the purpose of marking b}^ its edges the form of the 
face-mould, it will be observed that the face-mould is under- 
stood to have the position indicated by the hne ti^ b^^^, or at 



FACE-MOULDS FOR CIRCULAR STAIRS. 285 

the middle of the thickness of the plank. By this marking 
the rail-piece is cut square through the plank, and this cut- 
ting gives the correct form of the wreath, but only at the 
middle of the thickness of the plank. After it is cut square 
through the plank, then, to obtain the form at the upper and 
under surfaces, the face-mould is required to be moved end- 
wise, but parallel with the auxihary plane B, and so far as to 
bring the face-mould into a position vertically over or under 
its true position at the middle of the thickness of the plank. 
For example, the point ^,,,,, if the mould were placed at the 
middle of the thickness of the plank, would be at the height 
of the point h^^^ ; but when upon the top of the plank, the 
point b^^^^ would have to be at the height of the point c^, 
therefore the mould must be so moved that the point b^^^^ 
shall pass from b^ to <: ; consequently b^, c is the distance 
the mould must be moved, or, as it is technically termed, 
the sliding distance; hence b^^^^ c^, which is equal to b^c, is 
the distance the mould is to be moved : up when on top, 
and down when underneath. This is more ^uUy explained 
in Art. 284. 

278. — Face-Mowlds for Circular Stairs. — At Fig. 160 so 
much of the horizontal projection of the hand-railing of 
stairs in Fig. 158 is repeated as extends from the joint b to 
that at d, but at an enlarged scale. Upon the tangent c k 
set up the heights as given in Fig. 158 ; for example, make 
kk^ equal to k^^^k^^ of Fig. 158, and cc^ equal to c^^c^ of Fig. 
158. Join c^ and k^ and extend the line to meet ck, extended, 
in a. Join a and b ; then ab \^ the line of intersection of 
the cutting and horizontal planes ; it is therefore a horizon- 
tal line, parallel to which the ordinates are to be drawn. 
Perpendicular \,o ab draw b^c^^^^. Parallel to <^^ draw cc^^ 
and kk^i ; join b^ and c^^ ; the angle c c^^ b^ is the plumb-bevil ; 
perpendicular to b^ c^. draw b^ b^^, k^^ k^^^, and c^^ c^^^ ; make b, b^^ 
equal to b^ b, and so of the other two points, k^^^ and c.^^, make 
them respectively equal to their horizontal projections upon 
the plane A. Join c^.^ and k^/, also, k^^ and b^/, then b^.k.^^ 
and k^^^Cj^^ are the tangents. From c^^^ draw the line c^^J)^^ 
parallel to b^c^^ ; this is the slide-line. In this example, this 



286 



STAIRS. 



line passes through the point b,/, the sHde-line does not 
always pass through the ends of the two tangents; it is not 
required to pass through both, but it is indispensable that it 
be drawn parallel with the rake- line b^c^^. The lines for the 
joints at each end are drawn square to the tangent lines. 
Points in the curves, as many as are desirable, are now to 
be found by ordinates as shown in the figure, and as before 




explained for the points in the tangents. The curves are 
made by drawing a line against the side of a flexible strip 
bent to coincide with the points. 

The face-mould may be put in position by revolving the 
planes C and B, as explained in the last article, for the rail 
at the newel. 

The face-mould for the rail over the plan from c to d is to 



FACE-MOULDS CONTINUED. 287 

be obtained in a similar manner, taking the heights from Fig. 
158. For example, make d d^ equal to di^d^ of Ftg. 158, and 
11^ equal to Z.^, /^^ of Fig. 158 (taking the heights at their 
actual measurement now). Join d^ and /;, and extend the 
line to meet the line dl extended in r ; join r and c ; then re 
is the line of intersection, and parallel to which the ordinates 
are to be drawn. The points in the face-mould may now be 
obtained as in the previous cases, giving attention first to 
the tangent and slide-line ; drawing the lines for the joints 
perpendicular to the tangents. 

It may be remarked here that the chord-line be is parallel 
with the measuring line bjC^^^^y and that the line ^ X' bisects 
the chord-line; so, also, the line ^/bisects the chord-line cd. 
This coincidence is not accidental ; it will always occur in a 
regular circular stairs. 

Hence in cases of this kind it is not necessary to go 
through the preliminaries by which to obtain the intersect- 
ing line ab, but draw it at once parallel to the line ok, 
bisecting the chord be and passing through the point of 
intersection of the two tangents. For the distance to slide 
the mould in its after-application, the lines are given at e^^ 
and dj^j and their use is explained in the last article, and 
more fully in Art. 284. 

279.— Face-Moulds for Circular Stairs, again. — At Fig. 
161 so much of the plan of the hand-railing of the stairs of 
Fig. 158 is repeated as is required to show the rail from / 
to^, but drawn at a larger scale. To prepare for the face- 
moulds, perpendicular to // draw pp^, and make pp^ equal 
^^ PujPu o^ ^^^' 15^ (taking this height now at its actual 
measurement); join p^ and /; then //^ is the tangent of the 
vertical plane C, and / is a point in the cutting plane at its 
intersection with the base-plane A. Now since r s, the tan- 
gent over pg, is horizontal and is in the cutting plane, 
therefore from / draw fa parallel with r s ov pg\ then fa 
is the line of intersection of the cutting and horizontal 
planes, and gives direction to the ordinates. Draw f,p.^, 
perpendicular to fa ; make Pmp,, equal to pp^ ; join p^^ and 
/^ ; then the angle pp.^f, is the plumb-bevil ; perpendicular 



288 



STAIRS. 



to A,/, draw X/, and /,,/,,,, ; make p^J^^^^ equal to p^^^g, 
Pn^ equal to /,,,/; join d and /, ; then ^/^ and dp^^^, are 
the tangents. Make /,, ^ equal to half the thickness of the 
plank ; draw /, a parallel with //^^ ; make /, a equal to ^<: ; 
draw aCj parallel with the tangent/,^; through /,,, per- 
pendicular to /,, d, draw the line for the butt-joint ; then f^^c^ 
is the distance required to determine the vertical line on the 
face of the joint at f^^, as shown at A, Through /^^^^, per- 




pendicular to the tangent p^^,^ d, draw the line for the butt- 
joint ; make /,,,, <^ equal to ^^; then p^^^^b is the distance 
required for determining the vertical line on the face of the 
joint at /,,^^, as shown at B (see Art. 284). The curved lines 
are obtained by drawing a line against the edges of a flexi- 
ble rod bent to as many points as desirable, obtained by 
measuring the ordinates of the plan at A and transferring 
them to the face-mould by the corresponding ordinates, as 
before explained. 



RAILING FOR QUARTER-CIRCLE STAIRS. 289 

280. — Hand-Railings for ^Vinding^ Stair§. — The term 
winding is applied more particularly to a stairs having steps 
of parallel width compounded with those which taper in 
w^idth, as in Fig. 135, and as is here shown in Fig. 162, in 
which f abc represents the central line of the rail around the 
cylinder, and the quadrant dc, distant from the first quadrant 
20 inches, is the tread-line, upon which from d, a point taken 
at pleasure, the treads are run off. Through c, perpendicu- 




lar to af, draw ae (the occurrence here of one of the 
points of division on the tread-line perpendicularly opposite 
a, the spring of the circle, is only an accidental coincidence) ; 
make a a^ equal to two risers ; join a^ and /. With the 
diameter <^r, on ^ as a centre, describe the arc at g, crossing 
ac extended; through /; draw gb^\ then ab^ is the stretch- 
out, or development of the quadrant a b. 

Through Ji draw // /, tending toward the centre of the 



290 STAIRS. 

cylinder; make b^ i^ equal to bi\ perpendicular to fh^ draw 
b^ b^^ and i^ i^^. As there are four risers from e to Ji, make 
a^ a^^ equal to four risers, and draw a^, i^^ parallel with fa ; 
through / ^ draw a^ b^^ ; by intersecting lines, or in any con- 
venient manner, ease off to any extent the angle fa^ i^^. 
Through j\ a point in this curve (chosen so as to be perpen- 
dicularly over in, a point between a and /, nearer to a), 
draw k ly a tangent to the curve. Perpendicularly to this 
tangent, through j\ draw the line for a butt-joint ; also 
through bj^, and perpendicularly to a^ b^^, draw the line for 
the joint at the centre of the half circle. On the line aa^^ 
set up points of division for the riser heights, and through 
these points of division draw horizontal lines to the line 

b,Jf. 

From these points of contact drop perpendiculars to the 
line fa b^, and transfer such of them as require it to the circle 
ai, by drawing lines tending to^. Through these points of 
intersection with the central line of the rail, and through the 
points of division on the tread-line, draw the riser-lines uie, 
an, etc. At half a riser above the floor-line, on top of the 
upper riser draw a horizontal line, and ease off the angle as 
shown ; the intersection of the floor-line "with this curve 
gives the position of the top riser at the centre of the rail. 
This completes the plan of the steps and the elevation of the 
rail — requisite preliminaries for the face-moulds. The gradu- 
ation of the treads from flyers to winders obviates an abrupt 
angle at their junction in the rail and front-string. The 
objection to the graduation, that it interferes with the 
regularity of stepping at the tread-line, is not realized in 
practice. 

281. — Face-M®Mlds for \¥indiiig Stairs. — At Fig. 163 so 
much of the plan at Fig. 162 is repeated as is required for the 
face-moulds, but for perspicuity at twice the size. The hori- 
zontal projection of the tangents for the first wreath are ad 
and d b drawn at right angles to each other, tangent to the 
circle at a and b. Let those tangents be extended beyond 
d\ through ;//, the lower end of the wreath, draw nt d^, mak- 
ing an angle with nid equal to that in Fig. 162, between the 



FACE-MOULDS FOR THE TWISTS. 



291 



line af and aj\ or let the angle dmd^ equal afa^ of Fig. 
162. Make dd,^ equal to dd^. Make bh^^ equal to ^,,, <^,, of 
Fig. 162 ; join ^^^ and b^^ and extend the line to e^^ ; make 
b^, equal to ^,,<^,,,, of Fig 162, and draw <^v^^, parallel with 




de. From r^ draw e^^e parallel with b^^b\ through e and / 
draw ef tangent to the circle at /; then be and ef are the 
horizontal projections of the tangents for the upper wreath. 
Then if the plane B be revolved on ad, the plane C on de, 



292 STAIRS. 

and the plane D on cf until they each stand vertical to the 
plane A, the lines ind^, d^^e^^, and e^.J^ will constitute the 
tangents of the two wreaths in position. This arrangement 
locates the upper joint of the vipper wreath at /, leaving /<:, 
a part of the circle, to be worked as a part of the long level 
rail on the landing. As the tangent over e f is level, the 
raking part of the rail will all be included in the wreath bf, 
so that at the joint / the rail terminates on the level. 

The portion fc, therefore, is a level rail requiring no 
canting, and it requires no other face-mould than that afforded 
by the plan from / to c. 

For the face-mould for the rail over vi a b, let the line c^^ d^^ 
be extended to m^, a point in the base-line b m^ ; then in^ is a 
point in the base-plane A, as well as in the cutting plane E\ 
therefore the line m^ 111 is the intersecting line parallel to 
which all the ordinates on plane A are to be drawn. Per- 
pendicular to this intersecting line in^. in, at any convenient 
place draw in, b^ ; make b^ b^^^ parallel to ;//,, in and equal to 
bbjj] connect b^^, with m^, a point at the intersection of the 
lines m^ in and b^ in^ ; then the angle b b^^^ in^ is the plumb- 
bevil. Through d, parallel to m^ in, dmw d d^^/, from the 
three points in^, d^^^, and Z'^.^ draw lines perpendicular to 
in. b^/j ; make in^ in^^ equal to in^ in ; make b^^^ b^^^^ equal to b^ b. 
Since the measuring base-line in^ b^ passes through d, the 
point of the angle formed by the two tangents, d^^^ is the 
point of this angle in the cutting plane E\ therefore join in^^ 
and d^^^, also d^^^ and b^^^^ ; then b^^^^ d^^^ and d^^^ m^, are the 
two tangents at right angles to which the joints at in^^ and 
b^i^i are drawn. The curves of the face-mould are now found 
as usual, by transferring the distances by ordinates, as shown, 
from the plane A to the plane E, making the distance from 
the rake-line w b^^^ to each point in plane E equal to the dis- 
tance from the corresponding point in the plane A to the 
measuring base-line in^ b^. Now, to obtain the sliding distance 
and the vertical line upon the butt-joints, make b^^^ b,. equal 
to half the thickness of the plank; parallel with in^ b^^^ draw 
b^ /'vi ; ^Iso, b^^^^ ^vii and in^^ m^^^ ; make b^^^^ <^vu and in^^ m^^^ 
each equal to b^ b^-^\ through /;,.h and ni^^^, and parallel to the 
respective tangents, draw /Vu ^x and m^.^ ^^^,^1'^ t^"'^" ^^ and 



FACE-MOULDS FOR WIND^iNG STAIRS. 293 

m^^^t are the points from which, through the centre of the 
butt-joints, a line is to be drawn which will be vertical when 
the wreath is in position. (See Art. 284.) 

For the face-mould for the upper quarter, through b, Fig. 

163, draw b c^ parallel with d,^ c^^ ; make e c^^^ equal to e e^\ 
draw ^//^// parallel with e f. Now, since c^^^ f^ is a horizon- 
tal line and is in the cutting plane F, therefore, parallel with 
e^j^ f^ and through b^, draw b n ; then b n is the required in- 
tersecting line. Extend c f to/^^ ; make//^^ equal to f f^ ; 
join/^^ and n ; then the angle //^^ ;/ is the plumb-bevil. Per- 
pendicular to 11 f,^ draw/^^/^/ and n n^, and make these lines 
respectively equal to c f and b n ; join f^^ and f^^^ ; also f^^^ 
and n^ ; then f^^ f^^^ and f^^^ n^ are the required tangents. 
The butt-joints at/^ and n^ are drawn perpendicular to their 
respective tangents. To get the slide distance and vertical 
lines on the butt-joints, make/^^/^ equal to half the thickness 
of the plank ; parallel with n /^^, through /,, draw/,./^^^^ ; also, 
through n^ draw n^ n^^ ; make n^ n.^ equal to f^f^jj/, through 
;^,,, parallel with ;^^/^^^, draw ;/^^ n^^.] then ;/^^ , is the point 
through which a line is to be drawn to the centre of the 
butt-joint, and this line will be in the vertical plane contain- 
ing the tangent. So, also, parallel with the tangent f^^ /^^^, 
and through /^^^^, draw /^^^^ /,.i ; then/vj is the point through 
which a line is to be drawn to the centre of the butt-joint 
(see Art. 284). The curve is now to be obtained by the 
ordinates, as before explained. 

282. — Faec-lIould§ for Windaiag Slair§, ag^asoi. — In the 

last article, in getting the face-moulds for a w^inding stairs, 
the two wu-eaths are found to be very dissimilar in length. 
This dissimilarity may be obviated by a judicious location of 
the butt-joint connecting the two wreaths, as shown in Fig. 

164. Instead of locating the joint precisely at the middle of 
the half circle, as was done in Ftg. 163, place it farther down,' 
say at n, which is at it in Fig. 162, two risers down from the 
top, or at any other point at will. Then through 71 in the 
plan draw in^ s tangent to the circle at n ; and perpendicu- 
lar to this tano^ent draw n it,,, and d d,, ; make 11 n , equal to 
n^ It of Fig. 162 ; from d erect d d^ perpendicular to in d; 



294 



STAIRS. 



make the angle d m d^ equal to that of h^^^ j I of Fig. 162. 
Make d d^^ equal to d d, ; join d^^ and 71^^ and extend the line 
to m^^ a point of intersection with the base-line n n^ ; then 71^ 
is a point in the base-plane, as also in the cutting plane ; 




therefore 7?tj 7Ji is the intersecting line parallel to which all 
the ordinates of the plan are to be drawn, and perpendicular 
to which 771^^ n^, the measuring base-line, is drawn. Make 
Ttj 71^^^^ equal to 71 n^^ ; connect w^^ and n.^^^, and then transfer 



CARE REQUIRED IX DRAWIxXG. 295 

by the ordinates to the cutting" plane ;;/ d and ;/ the three 
points of the plan at the ends of the tangents, as before de- 
scribed, as also such points in the curve as may be required 
to mark the curve upon the face-mould, all as shown in previ- 
ous examples. For the face-mould of the upper wreath, make 
n^i n^ij equal to nn^^ of Fig. 162. From n^^^ draw n^^^ s^^ par- 
allel with in^ s ; extend the line d^. /r^ to intersect n^^^ 5^ in s^^ ; 
parallel with n^^^ n draw s.^ s ; from s draw s r tangent to the 
circle at r is n equals s r) ; through 7', tending to the centre of 
the cylinder, draw the butt-joint ; then r s and s n are the 
horizontal projections of the tangents for the upper wreath- 
piece, the tangent s r being level and, consequently, parallel 
to the intersecting line drawn through ;/. Perpendicular to 
r s draw r^p ; parallel with n^^ s^^ draw 71 s^ ; make r^ r ^ equal 
to s s,; join r^^ and /. From this line and the measuring base- 
line r^ p^ the points for the tangents are first to be obtained 
and then the points in the curve, all as before described. 
The part of the circle from r to <f is on the level, as before 
shown, and may be worked upon the end of the long level 
rail, its form being just what is show^n in the plan from c to r. 

283. — Face-MotBBc8§ : Test of Accuracy. — The methods 
which have been advanced for obtaining face-moulds are 
based upon principles of such undoubted correctness that 
there can be no question as to the results, when the methods 
given are thoroughly followed. And yet, notwithstanding 
the correctness of the system and its thorough comprehen- 
sion by the stair-builder, he wull fail of success unless he 
exercises the greatest care in getting his dimensions, his per- 
pendiculars, and his angles. The slightest deviation in a 
perpendicular terminated by an oblique line will result in a 
magnified error at the oblique line. To secure the greatest 
possible degree of accuracy, care must be exercised in the 
choice of the instruments by which the drawings are to be 
made : care to know that a straight-edge is what it purports 
to be; that a square, or right-angle, is truly a right-angle; 
that the compasses or dividers be well made, the joint per- 
fect, and the ends neatly ground to a point. Then let the 
drawing-board be carefully planed to a true surface ; and, 



296 STAIRS. 

if possible, let the drawing, full size, be made upon large, 
stout roll-paper rather than upon the drawing-board itself, 
as then the points for the face-mould may be pricked through 
upon the board out of which the face-mould is to be cut, and 
thus a correct transfer be made. For long straight lines it 
is better to use a fine chalk-line than the edge of a wooden 
straight-edge. The line is more trustworthy. Perpendicu- 
lars, especially Avhen long, are better obtained by measure- 
ment or by calculation {Art, 503) than by a square. The 
pencil used should be of fine quality — rather hard, in order 
that its point may be kept fine. With these precautions in 
regard to the instruments used, and with due care in the 
manipulations, the face-moulds may be correctly drawn, 
accurate in size and form. As a test of the accuracy of the 
work, it wnll be well to observe in regard to the tangents, 
that the length of a tangent, as found upon the face-mould, 
should always equal its length as shown upon the vertical 
plane. .For example, in Fig. 160, the tangent /^^^ c.^^ on the 
face-mould should be equal to /^^ c^, the tangent on the vertical 
plane B ; and in cases like this, where the stairs are quite 
regular, with equal treads at the front-string, the two tan- 
gents of a face-mould are equal to each other, or k^^ c^^ equals 
k^j b^j ; and in this case, the line b^^ c ^^^ should equal the rake- 
line b^ c^j. 

Again, as another example, in Fig. 161, d f^^, the tangent 
upon the face-mould, should be equal to//^, the tangent of 
the vertical plane C \ while d p^^^, the other tangent on the 
face-mould, should be equal to r s, the tangent of the vertical 
plane £>. But the more important test is in the length of the 
chord-line joining the ends of the tw^o tangents ; as, for ex- 
ample, the chord w^^ b^^^^ of Fig. 163, the horizontal projec- 
tion of which is the chord 7n b in plane A. Perpendicular to 
m b draw b g\ make b g equal to b b^^, and join g and m ; then 
m^j ^////» the chord of the face-mould, should be equal to m g. 
After fully testing the accuracy of the drawing for the face- 
mould, choose a well-seasoned thin piece of white-wood, or 
any other wood not Hable to split, and plane it to an even 
thickness throughout ; mark upon it the curves, joints, tan- 
gents, and slide-hne, and cut the edges true to the curve- 



THE FACE-MOULD APPLIED TO PLANK. 297 

lines and joints square through the board ; then square over 
such marks as are required to draw each tangent and the 
sUde-hne also upon the reverse side of the board. This 
completes the face-mould. 

284. — Application of tlie Face-Mould. — In order that a 
more comprehensive idea of the lines given for applying a 
face-mould may be had, let A, Fig. 165, represent one end of 
a wreath-piece as it appears when first cut from a plank, and 
when held up in the position it is to occupy at completion 
over the stairs. Also, let B represent the corresponding 
face-mould, laid upon the wreath-piece A in the position 
which it should have after sliding. And, for the purpose of 
a clearer illustration, let it be supposed that the two pieces, 
A and B, are transparent. Then let a^ a b d Cj e, represent a 
solid of wedge form, having a triangular level base, a b d, 
upon the three lines of which stand these three vertical 
planes, namely: on the line a b the plane a^ a bc^, upon the 
line a d the plane a^ a d e^, and on the line d b the plane b d 
c^ c^ ; the top of the solid is an inclined plane, a^ c^ e^, and the 
vertical line a^ a is the edge of the wedge. Now, it will be 
observed that the point a in the base of the solid is identical 
with a, the centre of the butt-joint, and the point a^ (at the 
intersection of two vertical planes and the inclined plane of 
the solid) is vertically over a, and is identical with a^, a poini 
in the upper surface of the plank. Also, the inclined plane 
c^ c^ a^, which forms the top of the solid, coincides with the 
upper surface of the plank A, from Avhich the Avreath-piece 
has been squared ; and the line c^ a^ (at the angle formed by 
the inclined plane e^ c^ a^ and the vertical plane a^ab c^i coin- 
cides with / g, the slide-line drawn upon the top of the 
plank ; also, the line e^ a^ (at the angle formed by the in- 
clined plane c^ c, a^ and the vertical plane a^ a d e), coincides 
with a^ k, the tangent line upon the underside of the face- 
mould after it has been slid to its new position, vertically 
over its true position at the middle of the thickness of the 
plank. From a the line a c is. drawn parallel with a^ c^ ; so, 
also, the line ^ ^ is drawn parallel with a^ c^ ; consequently 
the line ^ <; is parallel with e^ c^ ; and the plane e c a\^ parallel 



298 



STAIRS. 



with the plane c^ c^ a,, and coincides with a plane passing- 
through the middle of the thickness of the plank, and, conse- 
quently, is the cutting plane referred to in previous articles, 
upon which the lines are drawn which give shape to the 




Fig. i6- 



face-mould. When the face-mould is first laid upon the plank, 
the line i^j\ coincides with i^^ j\^, and when in that position, 
its form marked upon the plank is the form by which the 
plank is sawed square through ; but this gives the form of 



THE SLIDING OF THE FACE-MOULD. 299 

the wreath, not as it is at the surface of the plank, but as it 
is at the middle of the thickness of the plank, or upon the 
plane ac c\ so that, for example, the line inj\, represents the 
line ij drawn through a^ the centre of the butt-joint ; and 
when the mould B is slid to the position shown in the figure, 
the line ij\ comes into a position vertically over ij\ hence 
the three lines i^ i, a^ a, and j j' are each vertical and in a 
vertical plane, i i^jj- .By these considerations it will be seen 
that the face-mould By located as shown in the figure, is in its 
true position for the second marking, by which the addi- 
tional cutting is now to be performed vertically. This being 
established, it will now be shown how to get upon the butt- 
joint a line in the vertical plane containing the tangent. If 
the top and bottom lines of the vertical plane a^a b c^ be ex- 
tended, they will meet in the point /, and will extend the 
plane into a triangle lb c^, cutting the upper edge of the 
butt-joint in/, the end of the tangent, and the point in which 
the point a^ of the underside of the face-mould was located 
when the mould was first applied to the plank. The \\\\^ fa 
on the butt-joint is perpendicular to i j ox i^^ j,i> Again, if 
the top and bottom lines of the plane a^ a d c^ be extended, 
they will meet in /, and w411 extend the plane into the tri- 
angle p d e^, cutting the edge of the butt-joint in h, a point 
from which, if a line be drawn upon the butt-joint to a, its 
centre, this line will be in the vertical plane / d c^, which 
plane contains the tangent perpendicular to which the butt- 
joint is drawn ; consequently lines upon the butt-joint par- 
allel to Ji a will each be in a vertical plane parallel to the 
vertical tangent plane, and lines drawn upon the butt-joint 
perpendicular to these lines will be horizontal lines ; hence 
the line Jl a is the required line by which to square the 
wreath at the butt-joint. Now, it will be observed that the 
triangle a f a^ is like that given in the various figures for ob- 
taining face-moulds, to regulate the sliding of the face-mould 
and the squaring at the butt-joint. For example, in Fig. 
163, the right-angled triangle b^^^ b^ b^^ is the one referred to. 
This triangle is in a vertical plane pgirallel to one containing 
the slide-line ; its longer side is a vertical line ; one of the 
sides containing the right angle is equal to half the thickness 



300 



STAIRS. 



of the plank, while the other, drawn parallel to the face of 
the plank, is the distance the face-mould is required to slide. 
Precisely like this, the triangle a f a^ of Fig. 165 is in the 
vertical plane / b c^, containing f g, the slide-line ; its longer 
side, a J a, is a vertical line ; fa, one of the sides containing 
the right angle, is equal to half the thickness of the plank, 
while the other side, drawn coincident with the surface of 




Fig. 166. 



the plank, is the distance to slide the face-mould. Therefore 
the triangle a^f a of Fig. 165 gives the required lines by 
which to regulate the application of the face-moulds. The 
relative position of the more important of these lines is geo- 
metrically shown in Fig. 166, in which A and B are upon the 
horizontal plane of the paper, C is in a vertical plane stand- 
ing on the ground-line b d, and i? is a plan of the butt-joint, 
revolved upon the line i^^ j\^ into the horizontal plane, and 



BLOCKING-OUT OF THE RAIL. 30I 

then perpendicularly removed to the distance//^. The let- 
tering- corresponds with that in Fig. 165. The shaded part 
of D shows the end of the squared wreath. When the 
blocked piece has been marked by the face-mould in its 
second application, its edges are to be trimmed vertically as 
shown in Fig. 167, after which the top and bottom surfaces 
of the wreath are to be formed from the shape marked 6n 
the butt-joints. 




Fig. 167. 

285. — Face-Mould Curves are Elliptical. — The curves of 
the face-mould for the hand-railing of any stairs of circular 
plan are elliptical, and may be drawn by a trammel, or 
in any other convenient manner. The trouble, however, 
attending the process of obtaining the axes, so as to be able 
to employ the trammel in describing the curves, is, in many 
cases, greater than it would be to obtain the curves through 
points found by ordinates, in the usual manner. But as 



302 



STAIRS. 



this method for certain reasons may be preferred by some, 
an example is here given in which the curves are drawn by 
a trammel, and in Avhich the method of obtaining the axes is 
shown. 

Let Fig. i68 represent the plan of a hand-rail around part 




Fig. i68. 



of a cylinder and with the heights set up, the intersection 
line obtained, the measuring base-Hne drawn, the rake-line 
estabhshed, and the tangents on the face-mould located — all 
in the usual manner as hereinbefore described. Then, to 
prepare for the trammel, from o, the centre of the cylinder, 
draw o b^ parallel with the intersecting line, and b^ o^ perpen- 



FACE-MOULDS FOR ROUND RAILS. 303 

diciilar to l\f^, the rake-line ; make b^o^ equal to bo, and o^a^ 
equal \.o oa\ through 0^ draw o^Ji parallel with b^f^. From 
draw oc perpendicular to ob/, continue the central circular 
line of the rail around to e\ parallel with ob^ draw c f, 
and from /^, the point of intersection of e f with b^f^, and 
perpendicular to bj^, draw / r^ ; make f^L\ equal to fc\ 
then o^ is the centre of the ellipse, and o^ a^ the semi-conju- 
gate diameter and 0^ e^ the semi-transverse diameter of an 
ellipse drawn through the centre of the face-mould. To get 
the diameters for the edges of the face-mould, make a^ c^ and 
a^d^ each equal to half the widtii of the rail, as at cad\ par- 
allel to a line drawn from a^ to e^^ and through c^, draw the 
line c^g\ also, parallel with a line drawn from a^ to e^ draw 
dji (see Art. 559); then for the curve at the inner edge of 
the face-mould, o^g is the semi-transverse diameter, and o^c^ 
the semi-conjugate ; while for the curve at the outer edge 
0^ h is the semi-transverse diameter, and o^ d^ the semi-conju- 
gate. So much of the edges of the face-mould as are straight 
are parallel with the tangent. Now, placing the trammel at 
the centre, as shown in the figure, and making the distance 
on the rod from the pencil to the first pin equal to the 
semi-conjugate diameter, and the distance to the second pin 
equal to the semi-transverse diameter, each curve may be 
drawn as shown. (See Art. 549.) 

286.— Face-Moiiltls for Roaiiid Rails. — The previous ex- 
amples given for finding face-moulds are intended for moiddcd 
rails. For round rails the same process is to be followed, 
with this difference : that instead of finding curves on the 
face-mould for the sides of the rail, find one for a centre-line 
and describe circles upon it, as at Fig. 145 ; then trace the 
sides of the mould by the points so found. The thickness of 
stuff for the twists of a round rail is the same as for the 
straight part. The tw^ists are to be sawed square through. 

287. — Posiiion of llie Butt-Joint. — When a block for 
the wreath of a hand-rail is sawed square through the 
plank, the joint, in all cases, is to be laid on the face-mould 
square to the tangent and cut square through the plank. 



304 



STAIRS, 



Managed in this way, the butt-joint is in a plane pierced 
perpendicularly by the tangent. But if the block be not 
sawed square through, but vertically from the edges of the 




Fig. 169, 

face-mould, then, especially, care is required in locating the 
joint. The method of sawing square through is attended 
with so many advantages that it is now generally followed ; 
yet, as it is possible that for certain reasons some may prefer, 



POSITION OF THE BUTT-JOINT. 305 

in some cases, to saw vertically, it is proper that the method 
of finding the position of the joint for that purpose should 
be given. Therefore, let A, Fig. 169, be the plan of the rail, 
and B the elevation, showing its side; in which kz is the 
direction of the butt-joint. From /' draw kb parallel to lo, 
and ke at right angles to kb\ from h draw b f, tending to 
the centre of the plan, and from / draw f e parallel to bk\ 
from /, through r, draw I i, and from i draw id parallel to 
e f\ join ^and b, and db will be the proper direction for the 
joint on the plan. The direction of the joint on the other 
side, ac^ can be found by transferring the distances xb and 
d to xa and oc. Then the allowance for over wood to 
cover the butt-joint is shown as that which is included be- 
tween the lines ox and db. The face-mould must be so 
drawn as to cover the plan to the line b d for the wreath at 
the left, and to the line ac for that at the right. By some 
the direction of the joint is made to radiate toward the 
centre of the cylinder ; indeed, even Mr. Nicholson, in his 
Carpenter s Guide, so advised. That this is an error may be 
shown as follows : In Fig. 170, a rji is the plan of a part of the 
rail about the joint, s u is the stretch-out oi a /, and gp is the 
helinet, or vertical projection of the plan arji. This is 
found by drawing a horizontal line from the height set upon 
each perpendicular standing upon the stretch-out line S7i. 
The lines upon the plan arji are drawn radiating to the 
centre of the cylinder, and therefore correspond to the 
horizontal lines of the helinet drawn upon its upper and 
under surfaces. 

Bisect r^ on the ordinate drawn from the centre of the 
plan, and through the middle draw cb at right angles to gv ; 
from b and c draw cd and ^^ at risfht ans^les to j // ; from d 
and e draw lines radiating toward the centre of the plan ; 
then do and em will be the direction of the joint on the 
plan, according to Nicholson, and eb its direction on the 
falling-mould. It must be admitted that all the lines on the 
upper or the lower side of the rail w^hich radiate toward 
the centre of the cylinder, as do^ em, or if., are level; for 
instance, the level line w v, on the top of the rail in the 
helinet, is a true representation of the radiating line 72 on 



3o6 



STAIRS. 



the plan. The line bh, therefore, on the top of the rail in 
the helinet, is a true representation of cm on the plan, and 
kc on the bottom of the rail truly represents ^c. From k 
draw /^/parallel to cb, and from h draw Jif parallel \.obc\ 




Fig. 170. 

join / and b, also c and /; then cklb will be a true repre- 
sentation of the end of the lower piece, B, and cfhb of the 
end of the upper piece, A ; and fk or JlI will show how 
much the joint is open on the inner, or concave, side of the 
rail. 



CORRECT LINES FOR BUTT-JOINT. 



307 



To show that the process followed in Art. 287 is correct, 
let do and em {Fig. 171) be the direction of the butt-joint 
found as at Fig. i6g. Now, to project, on the top of the rail 
in the helinet, a line that does not radiate toward the centre 




of the cylinder, as jk, draw vertical lines from j and /c to w 
and /i, and join zu and /i ; then it will be evident that 7v /i is 
a true representation in the helinet of j7^ on the plan, it 
being in the same plane as j'k, and also in the same winding- 
surface as za2'. The line /;/, also, is a true representation on 



3o8 



STAIRS. 



the bottom of the helinet of the line j k in the plan. The 
line of the joint c in, therefore, is projected in the same way, 
and truly", by ib on the top of the helinet, and the line do 
by c a oxi the bottom. Join a and /, and then it will be seen 
that the lines c a,a i, and ib exactly coincide with c b, the line 
of the joint on the convex side of the rail ; thus proving the 
lower end of the upper piece. A, and the upper end of the 
lower piece, B, to be in one and the same plane, and that the 
direction of the joint on the plan is the true one. By refer- 
ence to Fig. 169 it will be seen that the line li corresponds 
to xi in Fig. 171 ; and that c k in that figure is a representa- 
tion oi fby and ik of db. 

288. — Scroll§ for IIanfl-Rail§ : Ocneral Rule for Size 
and Position of the Regulatings Square. — The breadth 
which the scroll is to occupy, the number of its revolutions, 
and the relative size of the regulating square to the eye of 
the scroll being given, multiply the number of revolutions 
by 4, and to the product add the number of times a side of 
the square is contained in the diameter of the eye, and the 
sum will be the number of equal parts into which the breadth 
is to be divided. Make a side of the regulating square 
equal to one of these parts. To the breadth of the scroll 
add one of the parts thus found, and half the sum will be the 
length of the longest ordinate. 



4 



Fig. 172. 



289. — Centres In Regulating Square. — Let a 2 \ b (Fig. 
172) be the size of a regulating square, found according to 
the previous rule, the required number of revolutions being 



SCROLLS AT NEWEL. 



309 



I J. Divide two adjacent sides, as a 2 and 2 i, into as many 
equal parts as there are quarters in the number of revolu- 
tions, as seven ; from those points of division draw lines 
across the square at right angles to the lines divided ; then 
I being the first centre, 2, 3,4, 5, 6, and 7 are the centres for 
the other quarters, and 8 is the centre for the eye ; the heavy 
lines that determine these centres being each one part less 
in length than its preceding line. 




Fig. 173, 



290, — Scroll for Hand-Rail Over Curtail Step. — Let 

a b {Fig. 173) be the given breadth, if the given number of 
revolutions, and let the relative size of the regulating square 
to the eye be \ of the diameter of the eye. Then, by the 
rule, if multipled by 4 gives 7, and 3, the number of times a 
side of the square is contained in the eye, being added, the 
sum is 10. Divide a b, therefore, into 10 equal parts, and set 
one from b \.o c \ bisect a c \n e-, then a c will be the length 
of the longest ordinate (i ^^ or i r). From a draw a d, 
from e draw e i, and from b draw b f, all at right angles to 
a b ; make e i equal to a, and through i draw i d parallel 



3IO STAIRS. 

to a b\ stt b c from i to 2, and upon i 2 complete the regu- 
lating square; divide this square as at Fig. 172; then de- 
scribe the arcs that compose the scroll, as follows : upon i 
describe d e, upon 2 describe e f, upon 3 describe / g, 
upon 4 describe ^/^, etc.; make dl equal to the width of the 
rail, and upon i describe / ;^/, upon 2 describe mn, etc.; de- 
scribe the eye upon 8, and the scroll is completed. 

291.— Scroll for Curtail Step.— Bisect d I {Fig. 173) in 0, 
and make v equal to ^ of the diameter of a baluster ; make 
V w equal to the projection of the nosing, and c x equal to 
zvl\ upon I describe w J, and upon 2 describe 7 -c ; also, upon 
2 describe x i, upon 3 describe ij\ and so around to z ; and 
the scroll for the step will be completed. 

292. — Po§iU©n of Balusters Under Scroll. — Bisect d I 
{Fig. 173) in 0, and upon i, with i for radius, describe the 
circle r ii \ set the baluster at p fair with the face of the 
second riser, <:", and from/, with half the tread in the divi- 
ders, space off as at 0, q, r, s, t, zCy etc., as far as ^- ; upon 2, 3,4, 
and 5 describe the centre-line of the rail around to the eye 
of the scroll ; from the points of division in the circle r 71 
draw lines to the centre-line of the rail, tending to 8, the 
centre of the eye ; then the intersection of these radiating 
lines Avith the centre-line of the rail will determine the posi- 
tion of the balusters, as shown in the figure. 

293. — Falling-Mould for Raking Part of Scroll. — Tangi- 
cal to the rail at h {Fig. 173) draw h k parallel to ^^^ ; then 
k a"- will be the joint between the twist and the other part 
of the scroll. Make d e'^ equal to the stretch-out of d c, and 
upon d c" find the position of the point /', as at k" \ at Fig. 
174, make ^<^ equal to <?- dm Fig. 173, and dc equal to d 0°^ 
in that figure ; from c draw c a Tit right angles to e c, and 
equal to one rise ; make c b equal to one tread, and from b, 
through ay draw b j\ bisect <^ ^ in /, and through / draw m q 
parallel to e h\ 111 q is the height of the level part of a 
scroll, which should always be about 3^ feet from the floor ; 
ease off the angle ni fj\ according to Art. 521, and draw 



FACE-MOULD FOR THE SCROLL. 



311 



£■ zv n parallel to m x j\ and at a distance equal to the thick- 
ness of the rail ; at a convenient place for the joint, as i, 
draw in at right angles to bj\ through 71 draw/ /^ at right 
angles to ^ // ; make dk equal to d k" in Fig. 173, and from k 
draw k o at right angles to e h ; at Fig. 173, make d //^ equal 
to d h in Fig. 174, and draw h- b"" at right angles to d h'^ -, 






..... 1.^;?^ 


T\ 


^^^d--^^ 


2 


1 


^,/^': 





c k t .. ^ d c u 

Fig. 174. 

then k a'^ and //^ ^- will be the position of the joints on the 
plan, and, at Fig. 174, op and in their position on the falling- 
mould ; and po in {Fig. 174) will be the required falling- 
mould which is to be bent upon the vertical surface from h'^ 
to k {Fig, 173). 




Fig. 175. 



294-. — Face-Tflould for the Scroll. — At Fig. 173, from k 
draw k r^ at right angles to r- d] at Fig. 172, make h r 
equal to h'^ r^ in Fig. 173, and from r draw r 5- at right 
angles tor^ ; from the intersection oi r s with the level line 
m q, through /, draw s t\ :xt Fig. 173, make Ji^ b^^ equal io q t 
in Fig. 172, and join Z^- and r^; from a"-, and from as many 



31 



STAIRS. 



Other points in the arcs, a'^ I and k d, as is thought neces- 
sary, draw ordinates to r'^ d at right angles to the latter; 
make r b (Fig. 175) equal in its length and in its divisions to 
the line r"- b^^ in Fig, 173 ; from r, ;/, 0, p, q, and / draw the 
lines r k, n d, a, p c, q f, and / <; at right angles to r b, and 
equal to r^^ k, ^^i--,/~ <^", etc., \n Fig, 173 ; through the points 
thus found trace the curves k I and a c, and complete the 
face-mould, as shown in the tigure. This mould is to be ap- 
plied to a square-edged plank, with the edge, / b parallel to 
the edge of the plank. The rake-lines upon the edge of the 
plank are to be made to correspond to the angle s t h in 
Fig, 174. The thickness of stuff required for this mould is 
shown at Fig. 174, between the lines s t and u v — u v being 
drawn parallel to s t. 




295- — Form of Nenrel-Cap from a Section of the Rail. 

— Draw a b {Fig. 176) through the widest part of the given 
section, and parallel to c d\ bisect a b in e, and through a, r, 
and b draw h /, f g^ and k j at right angles \.o a b\ at a con- 



BORING FOR BALUSTERS. 



313 



venient place on the line f g, as o, with a radius equal to 
half the width of the cap, describe the circle ijg\ make r I 
equal \.o e b or e a\ join / and j\ also / and i\ from the curve 
f b to the Hne / j draw as many ordinates as is thought 
necessary parallel to f g ; from the points at which these 
ordinates meet the line I j\ and upon the centre, 0, describe 
arcs in continuation to meet op ; from n t x, etc., draw n s, tii^ 
etc., parallel to f g\ make n s, t u, etc., equal to e f, wv, etc.; 
make x y, etc., equal to -c d, etc.; make ^ 2, ^ 3, etc., equal to 
n, t, etc.; make 2 4 equal to 71 s, and in this way find the 
length of the lines crossing in ; through the points thus 
found describe the section of the newel-cap as shown in 
the figure. 




Fig. 177. 



296.— Borings for Balu§ters in a Round Rail before it 
is Rounded. — Make the angle o c t {Fig. 177) equal to the 
angle c t ?it Fig. 144 ; upon c describe a circle with a 
radius equal to half the thickness of the rail ; draw the tan- 
gent b d parallel to / <r, and complete the rectangle e b d f, 
having sides tangical to the circle ; from c draw ^ <?: at right 
angles to c\ then, b d being the bottom of the rail, set a 
gauge from b to a, and run it the whole length of the stuff ; 
in boring, place the centre of the bit in the gauge-mark at a, 
and bore in the direction a c. To do this easily, make chucks 
as represented in the figure, the bottom edge, ^//, being par- 
allel to c, and having a place sawed out, as e f, to receive 
the rail. These being nailed to the bench, the rail will be 
held steadily in its proper place for boring vertically. The 
distance apart that the balusters require to be, on the under- 
side of the rail, is one half the length of the rakc-sidc of the 
pitch-board. 



314 



STAIRS. 



SPLAYED WORK. 



297. — The Bevel§ in Splayed Work. — The principles 
employed in finding the lines in stairs are nearly allied to 
those required to find the bevels for splayed work — such as 
hoppers, bread-trays, etc. A method by which these may be 




Fig. 178. 

obtained Avill, therefore, here be shown. In Fig. I'jZ, ab c\s> 
the angle at which the work is splayed, and b d, on the 
upper edge of the board, is at right angles to ^ ^ ; make the 
angle f g j equal \.o ab c, and from / draw f Ji parallel to 
e a ; from b draw ^ ^ at right angles X.O a b\ through draw 
ie parallel to c b^ and join c and d\ then the angle a e d will 
be the proper bevil for the ends from the inside, or k d e 
from the outside. If a mitre-joint is required, set f g, the 
thickness of the stuff on the level, from e to ;//, and join ;;/ 
and d\ then kdni will be the proper bevil for a mitre-joint. 
If the upper edge of the splayed work is to be bevelled, 
so as to be horizontal when the work is placed in its proper 
position, then f g j\ the same as ^ <^ r, will be the proper 
bevel for that purpose. Suppose, therefore, that a piece in- 
dicated by the lines k g, g f, and f h were taken off; then a 
line drawn upon the bevelled surface from d at right angles 
to k d would show the true position of the joint, because it 
would be in the direction of the board for the other side ; 
but a line so drawn would pass through the point o, thus 
proving the principle correct. So, if a line were drawn upon 
the bevelled surface from d at an angle of 45 degrees to k d, 
it would pass through the point n. 




/ Arf-LJi/. /^:!/iru. i}c/ 



VIEW IN THE ALHAMBRA. 



SECTION IV.— DOORS AND WINDOWS. 

DOORS. 

298- — General Requiremenls. — Among the architectural 
arrangements of an edifice, the door is by no means the least 
in importance ; and if properly constructed, it is not only 
an article of use, but also of ornament, adding materially to 
the regularity and elegance of the apartments. The dimen- 
sions and style of finish of a door should be in accordance 
with the size and style of the building, or the apartment 
for which it is designed. As regards the utility of doors, 
the principal door to a public building should be of suffi- 
cient width to admit of a free passage for a crowd of people ; 
while that of a private apartment will be wide enough if it 
permit one person to pass without being incommoded. Ex- 
perience has determined that the least width allowable for 
this is 2 feet 8 inches ; although doors leading to inferior 
and unimportant rooms may, if circumstances require it, be 
as narrow as 2 feet 6 inches ; and doors for closets, where an 
entrance is seldom required, may be but 2 feet wide. The 
width of the principal door to a public building may be 
from 6 to 12 feet, according to the size of the building ; and 
the width of doors for a dwelling may be from 2 feet 8 
inches to 3 feet 6 inches. If the importance of an apart- 
ment in a dwelling be such as to require a door of greater 
width than 3 feet 6 inches, the opening should be closed 
with two doors, or a door in two folds ; generally, in such 
cases, where the opening is from 5 to 8 feet, folding or slid- 
ing doors are adopted. As to the height of a door, it should 
in no case be less than about 6 feet 3 inches ; and generally 
not less than 6 feet 8 inches. 

299. — The Proportion between Width and Heig^ht: of 

single doors, for a dwelling, should be as 2 is to 5 ; and, for 
entrance-doors to public buildings, as i is to 2. If the 
width is given and the height required of a door for a 



i6 



DOORS AND WINDOWS. 



dwelling, multiply the width by 5, and divide the product 
by 2 ; but if the height is given and the width required, 
divide by 5 and multiply by 2. Where two or more doors 
of different widths show in the same room, it is well to pro- 
portion the dimensions of the more important by the above 
rule, and make the narrower doors of the same height as 
the wider ones ; as all the doors in a suit of apartments, 
except the folding or sliding doors, have the best appear- 
ance when of one height. The proportions for folding or 
sliding doors should be such that the width may be equal 
to f of the height ; yet this rule needs some qualification ; 
for if the width of the opening be greater than one half the 
width of the room, there will not be a sufficient space left 




Fig. 179. 

for opening the doors ; also, the height should be about one 
tenth greater than that of the adjacent single doors. 



300. — Panels. — Where doors have but two panels in 
width, let the stiles and muntins be each J- of the width ; or, 
whatever number of panels there may be, let the united 
widths of the stiles and the muntins, or the whole Avidth of 
the solid, be equal to | of the width of the door. Thus : in 
a door 35 inches wide, containing two panels in wndth, the 
stiles should be 5 inches wide ; and in a door 3 feet 6 inches 
wide, the stiles should be 6 inches. If a door 3 feet 6 
inches wide is to have 3 panels in width, the stiles and 
muntins should be each 4^ inches wide, each panel being 8 
inches. The bottom rail and the lock-rail ought to be each 
equal in width to j^ of the height of the door ; and the top 



TRIMMINGS FOR DOORS. 31/ 

rail, and all others, of the same width as the stiles. The 
moulding on the panel should be equal in width to ^ of the 
width of the stile. 

301. — Trimiiiing^§. — Fig. 179 shows a method of trimming 
doors : a is the door-stud ; b, the lath and plaster ; r, the 
ground; d, the jamb ; e, the stop ; f and g, architrave casings ; 
and //, the door-stile. It is customary in ordinar}^ work to 
form the stop for the door by rebating the jamb. But when 
the door is thick and heavy, a better plan is to nail on a 
piece as at e in the figure. This piece can be fitted to the 
door and put on after the door is hung ; so, should the door 
be a trifle winding, this will correct the evil, and the door be 
made to shut solid. 

Fig. 180 is an elevation of a door and trimmings suitable 
for the best rooms of a dwelling. (For trimmings generally, 
see Sect. V.) The number of panels into which a door 
should be divided may be fixed at pleasure ; yet the present 
style of finishing requires that the number be as small as a 
proper regard for strength will admit. In some of our best 
dwellings, doors have been made having only two upright 
panels. A few years' experience, however, has proved that 
the omission of the lock-rail is at the expense of the strength 
and durability of the door ; a four-panel door, therefore, is 
the best that can be made. 

302. — IIangin§^ I>oor$. — Doors should all be hung so as 
to open into the principal rooms ; and, in general, no door 
should be hung to open into the hall, or passage. As to the 
proper edge of the door on which to affix the hinges, no 
general rule can be assigned. 



WINDOWS. 

303. — Requirements for Lig^lit. — A window should be 
of such dimensions, and in such a position, as to admit a 
sufficiency of light to that part of the apartment for which 
it is designed. No definite rule for the size can well be 
given that will answer in all cases ; yet, as an approxima- 



DOORS AND WINDOWS. 



tion, the following has been used for general purposes. 
Multiply together the length and the breadth in feet of the 
apartment to be lighted; and the product by the height in 




feet; then the square root of this product will show the 
required number of square feet of glass. 

304. — Window-Frames. — For the size of window-frames, 
add 4^ inches to the width of the glass for their width, and 



WIDTH OF INSIDE SHUTTERS. 319 

6J inches to the height of the glass for their height. These 
give the dimensions, in the clear, of ordinary frames for 12- 
light windows ; the height being taken at the inside edge of 
the sill. In a brick wall, the width of the opening is 8 
inches more than the width of the glass — 4^ for the stiles of 
the sash, and 3^ for hanging stiles — and the height between 
the stone sill and lintel is about loj- inches more than the 
height of the glass, it being varied according to the thick- 
ness of the sill of the frame. 

305. — Iii§i<]c SBawtters. — Inside shutters folding into 
doxes require to have the box-shutter about one inch wider 
than the flap, in order that the flap may not interfere when 
both are folded into the box. The usual margin shown be- 
tween the face of the shutter when folded into the box and 
the quirk of the stop-bead, or edge of the casing, is half an 
inch ; and, in the usual method of letting the whole of the 
thickness of the butt hinge into the edge of the box-shutter, 
it is necessary to make allowance for the throw of the hinge. 
This may, in general, be estimated at \ of an inch at each 
hinging ; which being added to the margin, the entire width 
of the shutters will be \\ inches more than the width of the 
frame in the clear. Then, to ascertain the width of the box- 
shutter, add i^ inches to the width of the frame in the clear, 
between the pulley-stiles ; divide this product by 4, and add 
half an inch to the quotient, and the last product will be 
the required width. For example, suppose the window to 
have 3 lights in Avidth, 11 inches each. Then, 3 times 11 is 
33, and 4J added for the wood of the sash gives 37^ ; 37^ 
and li is 39, and 39 divided by 4 gives 9J ; to which add 
half an inch, and the result will be loj inches, the width 
required for the box-shutter. 

306. — Proportion: "Widtli and 9Icight. — In disposing 
and locating windows in the walls of a building, the rules of 
architectural taste require that they be of different heights 
in different stories, but generally of the same width. The 
windows of the upper stories should all range perpendicu- 
larly over those of the first, or principal, story ; and they 



320 DOORS AND WINDOWS. 

should be disposed so as to exhibit a balance of parts 
throughout the front of the building. To aid in this it is 
always proper to place the front door in the middle of the 
front of the building ; and, where the size of the house will 
admit of it, this plan should be adopted. (See the latter 
part of y^r/. 50.) The proportion that the height should 
bear to the Avidth may be, in accordance with general usage, 
as follows : 



he height of basement wind 


ows 


, li 


of the width. 




principal-Story 


'' 


-8 


il 


il u 


second-story 


il 


Ij 


il 


li u 


third-story 


il 


It 


" 


u a 


fourth-story 


11 


I* 


il 


(t ii 


attic-story 


11 


the 


same as the width. 



But, in determining the height of the windows for the 
several stories, it is necessary to take into consideration the 
height of the story in wdiich the window is to be placed. 
For, in addition to the height Irom the floor, which is gen- 
erally required to be from 28 to 30 inches, room is wanted 
above the head of the window for the window-trimming 
and the cornice of the room, besides some respectable space 
which there ought to be between these. 

307. — Circsaflar Meads. — Doors and windows usually ter- 
minate in a horizontal line at top. These require no special 
directions for their trimmings. But circular-headed doors 
and windows are more difficult of execution, and require 
some attention. If the jambs of a door or window be placed 
at right angles to the face of the wall, the edges of the so^'it, 
or surface of the head, would be straight, and its length be 
found by getting the stretch-out of the circle {Art. 524); 
but when the jambs are placed obliquely to the face of the 
wall, occasioned by the demand for light in an oblique 
direction, the form of the soffit Avill be obtained by the fol- 
lowing article ; as also when the face of the wall is circular, 
as shown in the succeeding figure. 



OBLIQUE SOFFITS OF WINDOWS. 



32 



308.— Form of Soffit for Circular Window-Heads. 

When the light is received in an oblique direction, let ad cd 
{Fig. 181) be the ground-plan of a given window, and efa a 
vertical section taken at right angles to the face of the jambs. 




Fig. 181. 



From a, through e, draw ag at right angles to ^^; obtain 
the sti-etch-out of efa, and make ^^ equal to it; divide eg 
and efa each into a like number of equal parts, and drop 
perpendiculars from the points of division in each ; from 
the points of intersection, i, 2, 3, etc., in the line ad, 



f 













' 




e 


/ 


^ 












7 


"T^ 




— , — 


— 




— 


1 


\^ 


■"^z 


"?^^ 












1 










■ ■ 











■"-"^U 


"^ 


1 


' 


^ 


2 


^ 


/ 


2 


' \ 








1 




— 


1 r 


^ 




^=^- 


- -1 - 


— 


-■rzz^V^^ 


^"^ 



Fig. 182. 



draw horizontal lines to meet corresponding perpendicu- 
lars from eg\ then those points of intersection will give the 
curve line dg, which will be the one required for the edge 
of the sofBt. The other edge, ch, is found in the same 
manner. 



322 DOORS AND WINDOWS. 

For the form of the sofht for circular window-heads, 
when the face of the wall is curved, let ah c d (Fig. 182) be 
the ground-plan of a given window, and e f a 7\. vertical sec- 
tion of the head taken at right angles to the face of the 
jambs. Proceed as in the foregoing article to obtain the 
line dg\ then that will be the curve required for the edge of 
the soffit, the other edge being found in the same manner. 

If the given vertical section be taken in a line with the 
face of the wall, instead of at right angles to the face of the 
jambs, place it upon the line cb {Fig. 181), and, having drawn 
ordinates at right angles to cb, transfer them to efa ; in this 
way a section at right angles to the jambs can be obtained. 






SECTION v.— MOULDINGS AND CORNICES. 

MOULDINGS. 

309. — ifloiildings : are so called because they are of the 
same determinate shape throughout their length, as though 
the whole had been cast in the same mould or form. The 
regular mouldings, as found in remains of classic architec- 
ture, are eight in number, and are known by the following 
names : 



Fig. 183. Annulet, band, cincture, fillet, listel or square. 



Fig. 184. Astragal or bead. 



) 



Torus or tore. 



Fig. 185. 



Pj^ jg^ Scotia, trochilus or mouth. 



Fig. 1S7. 



Ovolo, quarter-round or echinus. 



Cavetto, cove or hollow. 



Fig. 188. 



324 MOULDINGS AND CORNICES. 



Fig. 189. 



Cymatium, or cyma-recta. 

Ogee. 



I Inverted cymatium, or cyma-reversa. 

Fig. 190. 

Some of the terms are derived thus : Fillet, from the French 
word fil, thread. Astragal, from astragalos, a bone of the 
heel — or the curvature of the heel. Bead, because this 
moulding, when properly carved, resembles a string of beads. 
Torus, or tore, the Greek for rope, which it resembles when 
on the base of a column. Scotia, from skotia, darkness, be- 
cause of the strong shadow which its depth produces, and 
which is increased by the projection of the torus above it. 
Ovolo, from oviuii, an ^%^^ which this member resembles, 
when carved, as in the Ionic capital. Cavetto, from caviis, 
hollow. Cymatium, from kinnaton, a wave. 

3(0. — Cliai'aetcrl§tic§ of Mouldiii^§. — Neither of these 
mouldings is peculiar to any one of the orders of architect- 
ure ; and although each has its appropriate use, yet it is by 
no means confined to any certain position in an assemblage 
of mouldings. The use of the fillet is to bind the parts, as 
also that of the astragal and torus, which resemble ropes. 
The ovolo and cyma-reversa are strong at their upper ex- 
tremities, and are therefore used to support projecting parts 
above them. The cyma-recta and cavetto, being weak at 
their upper extremities, are not used as supporters, but are 
placed uppermost to cover and shelter the other parts. The 
scotia is introduced in the base of a column to separate the 
upper and lower torus, and to produce a pleasing variety 
and relief. The form of the bead and that of the torus is the 
same ; the reasons for giving distinct names to them are 
that the torus, in every order, is always considerabl}^ larger 
than the bead, and is placed among the base mouldings. 



GRECIAN MOULDINGS. 



325 



whereas the bead is never placed there, but on the capital or 
entablature ; the torus, also, is seldom carved, whereas the 
bead is ; and while the torus among the Greeks is frequently 
elliptical in its form, the bead retains its circular shape. While 
the scotia is the reverse of the torus, the cavetto is the re- 
verse of the ovolo, and the cyma-recta and cyma-reversa are 
combinations of the ovolo and cavetto. 




Fig. 191. 

The curves of mouldings, in Roman architecture, were 
most generally composed of parts of circles ; while those of 
the Greeks were almost always elliptical, or of some one of 
the conic sections, but rarely circular, except in the case of 
the bead, which Avas always, among both Greeks and Ro- 
mans, of the form of a semicircle. Sections of the cone af- 
ford a greater variety of forms than those of the sphere ; and 
perhaps this is one reason why the Grecian architecture so 



326 



MOULDINGS AND CORNICES. 



much excels the Roman. The quick turnings of the ovolo 
and cyma-reversa, in particular, when exposed to a bright 
sun, cause those narrow, well-defined streaks of light which 
give life and splendor to the whole. 

311. — A Profile: is an assemblage of essential parts and 
mouldings. That profile produces the happiest effect which 




d 


l^i 


a 



FiGo 192. 



Fig. 193. 



is composed of but few members, varied in form and size, 
and arranged so that the plane and the curved surfaces suc- 
ceed each other alternatelv. 



312. — The Grecian Torus and Scotia. — Join the extremi- 
ties a and b {Fig. 191), and from /, the given projection of 
the moulding, draw/^ at right angles to the fillets ; from b 





\>. 1 


n 


^ 




' 



Fig. 194. 




Fig. 195. 



draw bh at right angles to ab\ bisect abvac, join /and r, 
and upon c, with the radius cf, describe the ^yc fh, cutting 
bh in // ; through c draw de parallel with the fillets ; make 
dc and c c each equal to b h\ then de and a b will be conju- 
gate diameters of the required ellipse. To describe the 
curve by intersection of lines, proceed as directed at Art. 



THE GRECIAN ECHINUS. 



327 



551 and note ; by a trammel, see Art. 549 ; and to find the 
foci, in order to describe it with a string, see Art. 548. 

313. — The Grecian i:clnnu§. — Figs. 192 to 199 exhibit, va- 
riously modified, the Grecian ovolo, or echinus. Figs. 192 to 
196 are elliptical, a b and b c being given tangents to the curve ; 
parallel to which the semi-conjugate diameters, ad Siud dc, 








fd 


V 




/ 



Fig. 196. 



Fig. 197. 



are drawn. In Figs. 192 and 193 the lines a d Sind dc are semi- 
axes, the tangents, ab and be, being at right angles to each 
other. To draw the curve, see ^r/. 551. In Fig. 197 the 
curve is parabolical, and is drawn according to Art. 560. In 
Figs. 198 and 199 the curve is hyperbolical, being described 
according to Art. 561. The length of the transverse axis, a b. 





Fig. 198. 



Fig. 199. 



being taken at pleasure in order to flatten the curve, a b 
should be made short in proportion to ac, 

3(4 — The Grecian Cavetto.— In order to describe this, 
Figs.2Qo^Vid 201, having the height and projection given, 
see Art. 551. 

315.— The Grecian Cyma-Recta.— When the projection 
is more than the height, as at Fig. 202, make ^<^ equal to the 



328 



MOULDINGS AND CORNICES. 



height, and divide abed into four equal parallelograms ; then 
proceed as directed in note to ^r/. 551. When the projec- 
tion is less than the height, draw da {Fig, 203) at right angles 





Fig. 201. 



Fig. 200. 



to ^^; complete the rectangle, abcd\ divide this into four 
equal rectangles, and proceed according \.o Art. 551. 

316. — The CJreciaii Cyiiia-Rever§a. — When the projection 



1 




\Mf' 


Im 


WW 


V 


c 



■ 1 




1 


i 




■' 





Fig. 202. 



Fig. 203. 



is more than the height, as at Fig. 204, proceed as directed 
for the last figure ; the curve being the same as that, the 
position only being changed. When the projection is less 



1 


1 




^ 


J5^ 


- 


a 


^ 


#s 


^^ 



Fig. 204. 



d 
Fig. 205. 



than the height, draw a d (Fig. 205) at right angles to the 
fillet ; make a d equal to the projection of the moulding ; then 
proceed as directed for Fig. 202. 



FORMS OF ROMAN MOULDINGS. 



329 



317. — Roman Moulding^s : are composed of parts of circles, 
and have, therefore, less beauty of form than the Grecian. 
The bead and torus are of the form of the semicircle, and the 
scotia, also, in some instances ; but the latter is often composed 
of two quadrants, having different radii, as at Figs. 206 and 
207, which resemble the elliptical curve. The ovolo and ca- 





FiG. 206. 



Fig. 207. 



vetto are generally a quadrant, but often less. When they are 
less, as at Fig: 210, the centre is found thus : join the extrem- 
ities, a and d, and bisect ad in c; from c, and at right angles 
to a d, draw c d, cutting a level line drawn from aind; then d 
will be the centre. This moulding projects less than its 
height. When the projection is more than the height, as at 
Fig. 212, extend the line from c until it cuts a perpendicular 





Fig. 208. 



Fig. 209. 



drawn from <^, as at <^; and that will be the centre of the 
curve. In a similar manner, the centres are found for the 
mouldings at Figs. 207,211, 213, 216, 217, 218, and 219. The 
centres for the curves at Figs. 220 and 221 are found thus : 
bisect the line ^ ^ at ^ ; upon a, c and b successively, with a c 
or cb for radius, describe arcs intersecting at d and d\ then 
those intersections will be the centres. 



330 



MOULDINGS AND CORNICES. 










^ 


1 



Fig. 210. 



Fig. 211. 



a 


\ 1 




}>^ 




a 




Fig. 212. 



Fig. 213. 



ffl 



Fig. 214. 



Fig. 215 





Fig. 216. 



Fig. 217. 



FORMS OF MODERN MOULDINGS. 



331 



318. — Modern Iflouldiiigs : are represented in Figs. 222 
to 229. They have been quite extensively and successfully 
used in inside finishing. Fig. 222 is appropriate for a bed- 
moulding under a low projecting shelf, and is frequently 
used under mantel-shelves. The tangent i h is found thus : 
bisect the line ab at c, and b c 'sX d\ from d draw dc at 
right angles to ^ <^ ; from b draw bf parallel to ^<^; upon b. 





Fig. 219. 

with b d for radius, describe the arc d f\ divide this arc 
into 7 equal parts, and set one of the parts from s, the limit 
of the projection, to ; make h equal to e\ from //, through 
c, draw the tangent h i; divide b h, lie, c i, and ia each into 
a like number of equal parts, and draw the intersecting lines 
as directed at Art. ^21. If a bolder form is desired, draw 
the tangent, i h, nearer horizontal, and describe an elliptic 





Fig.^ 220. 

curve as shown in Figs. 191 and 224. Fig. 223 is much used 
on base, or skirting, of rooms, and in deep panelling. The 
curve is found in the same manner as that of Fig. 222. In 
this case, however, where the moulding has so little projec- 
tion in comparison with its height, the point e being found 
as in the last figure, h s may be made equal to s c, instead of 
{? ^ as in the last figure. Fig. 224 is appropriate for a crown 



MOULDINGS AND CORNICES. 




Fig. 223. 




Fig. -224. 



PLAIN MOULDINGS. 



333 



moulding of a cornice. In this figure the height and pro- 
jection are given; the direction of the diameter, ab, drawn 





Fig. 225. 



Fig. 226. 



through the middle of the diagonal, e f, is taken at pleasure ; 
and ^<; is parallel to ac. To find the length oi dc^ draw b h 




Fig. 227. 



Fig. 228. 



Fig. 229. 



at right angles \.o ab ; upon 0, with of for radius, describe 
the arc, f h, cutting b Ji in Ji ; then make oc and d each 



334 



MOULDINGS AND CORNICES. 



equal to bh'^ To draw the curve, see note \.oArt. 551. Figs, 
225 to 229 are peculiarly distinct from ancient mouldings, 
being composed principally of straight lines ; the few curves 
they possess are quite short and quick. 

Figs. 230 and 231 are designs for antce caps. The di- 
ameter of the antae is divided into 20 equal parts, and the 
height and projection of the members are regulated in ac- 
cordance with those parts, as denoted under H and P, height 
and projection. The projection is measured from the mid- 
dle of the antas. These will be found appropriate for por- 
ticos, doorways, mantelpieces, door and window trimmings, 



H.P. 



5 15 




4 
121 


J 


2 11 


1 


9 101 




T 


1 
10 


, 






Fig. 230. 



Fig. 231. 



etc. The height of the antae for mantelpieces should be 
from 5 to 6 diameters, having an entablature of from 2 to 
2\ diameters. This is a good proportion, it being similar to 
the Doric order. But for a portico these proportions are 



* The manner of ascertaining the length of the conjugate diameter, d c, in 
this figure, and also in Figs. 191, 241, and 242 is new, and is important in this 
application. It is founded upon well-known mathematical principles, viz. : All 
the parallelograms that may be circumscribed about an ellipsis are equal to 
one another, and consequently any one is equal to the rectangle of the two 
axes. And again : The sum of the squares of every pair of conjugate diame- 
ters is equal to the sum of the squares of the two axes. 



EAVE CORNICES. 



335 



much too heavy : an antae 1 5 diameters high and an entab- 
lature of 3 diameters will have a better appearance. 

CORNICES. 

319- — Designs for Cornices. — Figs. 232 to 240 are designs 
for eave cornices, and Figs. 241 and 242 are for stucco cor- 
nices for the inside finish of rooms. In some of these the 
projection of the uppermost member from the facia is 
divided into twenty equal parts, and the various members 



! i 


\ . . 1 


1' r 


> 


^ 




Fig. 232. 

are proportioned according to those parts, as figured under 
77 and P. 



320 -—Eave Cornices Proportioned to Height of Build- 
ing. — Draw the line ac {Fig. 243), and make be and ba each 
equal to 36 inches ; from <^ draw bd Tit right angles to ac, 
and equal in length to f of ac\ bisect b dm c, and from a, 
through e, draw af\ upon a, Avith ac for radius, describe the 
arc cf, and upon e, with cfiov radius, describe the arc/<^; 
divide the curve dfc, into 7 equal parts, as at 10, 20, 30, 
etc., and from these points of division draw lines to ^^ 



336 



MOULDINGS AND CORNICES. 




Fig. 233. 



7 




1 



ki/%}/j%/j^U^i^ 



JULJlEJlIUUIJUUUJU'JUUUlI 






Fig. 234, 



-EXAMPLES OF CORNICES. 



337 



7 




riG. 235. 



7 




Fig. 236, 



338 



MOULDINGS AND CORNICES. 




Fig. 237. 



H 


P. 










n 20 




5 

n 
11 

8 

~i 

25 


m 


f 
J 


1 


16J 


; 








...) ^ c ■"•••-.. 




) 













Fig. 238. 



VARIOUS DESIGNS OF CORNICES. 



339 



H. P. 



U20 



1 17 



3^31 







H 


2i 


^3- 



Fig. 239. 



H.P. 



3|20 



16 




nn 



^ 

UiLXtLMJ 



Fig. 240. 




U 



340 



MOULDINGS AND CORNICES 



H. P. 




OODOODOO 



Fig. 241. 



H. P 




Fic. 242. 



PROPORTION OF CORNICES. 



341 



parallel to db\ then the distance b i is the projection of a 
cornice for a building 10 feet high ; b 2, the projection at 20 
feet high ; b 3, the projection at 30 feet, etc. If the projec- 
tion of a cornice for a building 34 feet high is required, 
divide the arc between 30 and 40 into 10 equal parts, and 




from the fourth point from 30 draw a line to the base, b c, 
parallel with b d\ then the distance of the point at which 
that line cuts the base from b will be the projection re- 
quired. So proceed for a cornice of any height within 70 
feet. The above is based on the supposition that 36 inches 




Fig. 244. 

is the proper projection for a cornice 70 feet high. This, 
for general purposes, will be found correct ; still, the length 
of the line be may be varied to suit the judgment of those 
who think differently. 

Having obtained the projection of a cornice, divide it 
into 20 equal parts, and apportion the several members 



342 



MOULDINGS AND CORNICES. 



according to its destination — as is shown at Figs. 238, 239, 
and 240. 

321. — Cornice Proportioned to a g^iven Cornice. — Let 

the cornice at Fig. 244 be the given cornice. Upon any 
point in the lowest Hne of the lowest member, as at a, with the 
height of the required cornice for radius, describe an intersect- 
ing arc across the uppermost line, as at <^ ; join a and b ; 
then b i will be the perpendicular height of the upper fillet 
for the proposed cornice, i 2 the height of the crown mould- 
ing — and so of all the members requiring to be enlarged to 
the sizes indicated on this line. For the projection of the 




245- 



proposed cornice, draw ^ <^ at right angles to a b, and c d dit 
right angles to ^ ^ ; parallel with c d draw lines from each 
projection of the given cornice to the line ad\ then ^<^ will 
be the required projection for the proposed cornice, and the 
perpendicular lines falling upon e d will indicate the proper 
projection for the members. 

To proportion a cornice according to a larger given cor- 
nice, let A {Fig.. 245) be the given cornice. Extend ao to b, 
and draw ^ <^ at right angles to ^ ^ ; extend the horizontal 
lines of the cornice, A, until they touch o d] place the height 
of the proposed cornice from to e, and join y and e\ upon 
Oy with the projection of the given cornice, a, for radius, 



TO FIND THE ANGLE BRACKET. 



;43 



describe the quadrant ad\ from ^ draw <^^ parallel to/^; 
upon 0, with o b for radius, describe the quadrant b c ; then 
c will be the proper projection for the proposed cornice. 
Join a and c ; draw lines from the projection of the different 
members of the given cornice to « 6* parallel Vo od\ from 
these divisions on the line a o draw lines to the line o c 
parallel \.o ac\ from the divisions on the line of draw lines 
to the line oe parallel to the line/r; then the divisions on 
the lines o e and o c will indicate the proper height and pro- 
jection for the different members of the proposed cornice. 
In this process, we have assumed the height, o e, of the pro- 
posed cornice to be given ; but if the projection, o c, alone 




Fig. 247. 



be given, we can obtain the same result b}' a different pro- 
cess. Thus: upon d?, with (? <: for radius, describe the quad- 
rant cb\ upon 0, with a for radius, describe the quadrant 
ad\ join d and b\ from /draw fc parallel to db\ then oc 
will be the proper height for the proposed cornice, and the 
height and projection of the different members can be 
obtained by the above directions. By this problem, a cor- 
nice can be proportioned according to a smaller given one 
as well as to a larger ; but the method described in the pre- 
vious article is much more simple for that purpose. 



322. — Angle Bracket in a Built Cornice. 

246) be the wall of the 



-Let A {Fi^ 



building, and B the given bracket, 



344 



OULDINGS AND CORNICES. 



which, for the present purpose, is turned down horizontally. 
The angle-bracket, C, is obtained thus : through the ex- 
tremity, a, and parallel with the wall,/<^, draw the line a b ; 
make^<: equal af, and through c draw cb parallel with ed\ 
join d and b^ and from the several angular points in B draw 
ordinates to cut db'm i, 2, and 3 ; at those points erect lines 
perpendicular to db\ from h draw hg parallel to fa; take 




Fig. 248. 



the ordinates, 10,20, etc., at B, and transfer them to C, and 
the angle-bracket, C, will be defined. In the same manner, 
the angle-bracket for an internal cornice, or the angle-rib of 
a coved ceiling, or of groins, as at Fig. 247, can be found. 



323,— Raking Mouldings matched with Level Returns.— 

Let A {Fig. 248) be the given moulding, and A b the rake of 



CROWN MOULDING ON THE RAKE. 345 

the roof. Divide the curve of the given moulding into any 
number of parts, equal or unequal, as at i, 2, and 3 ; from 
these points draw horizontal lines to a perpendicular erected 
from c ; at any convenient place on the rake, as at B, draw 
a c ?X right angles to A b\ also from b draw the horizontal 
line b a-, place the thickness, d a^ of the moulding at A from 
b to <^, and from <^ draw the perpendicular line ae\ from 
the points i, 2, 3, at ^, draw lines to C parallel X.O A b \ 
make a \, a2, and a 3, at B, and at C, equal to rt; i, etc., at A ; 
through the points, i, 2, and 3, at B, trace the curve — this 
will be the proper form for the raking moulding. From i, 
2, and 3, at C, drop perpendiculars to the corresponding 
ordinates from i, 2, and 3, at y^ ; through the points of inter- 
section, trace the curve — this will be the proper form for the 
return at the top. 



PART II 



SECTION VI.— GEOMETRY. 



324. — Mathematics Essential. — In this and the following 
Sections, which will constitute Part II., there are treated of 
certain matters which may be considered as elementary. 
They are all very necessary to be understood and acquired 
by the builder, and are here compactly presented in a shape 
which, it is believed, will aid him in his studies, and at the 
same time prove to be a great convenience as a matter of 
reference. 

The many geometrical forms which enter into the 
composition of a building suggest a knowledge of Elemen- 
tary Geometry as essential to an intelligent comprehension 
of its plan and purpose. One of the prime requisites of a 
building is stability, a quality which depends upon a proper 
distribution of the material of which the building is con- 
structed ; hence a knowledge of the laws of pressure and 
the strength of materials is essential ; and as these are based 
upon the laws of proportion and are expressed more con- 
cisely in algebraic language, a knowledge of Proportion and 
of Algebra are likewise indispensable to a comprehensive 
understanding of the subject. There will be found in this 
work, however, only so much of these parts of mathematics 
as have been deemed of the most obvious utility in the 
Science of Building. For a more exhaustive treatment of 
the subjects named, the reader is referred to the many able 
works, readily accessible, which make these subjects their 
specialties. 

325.— Elementary Geometry. — In all reasoning defini- 
tions are necessary, in order to insure in the minds of the 



348 GEOMETRY. 

proponent and respondent identity of ideas. A corollary is 
an inference deduced from a previous course of reasoning. 
An axiom is a proposition evident at first sight. In the fol- 
lowing demonstrations there are many axioms taken for 
granted (such as, things equal to the same thing are equal to 
one another, etc.) ; these it was thought not necessary to 
introduce in form. 

326. — Definition. — If a straight line, as A B {Fig. 249), 
stand upon another straight line, as CD, so that the two 



C B D 

Fig. 249. 

angles made at the point B are equal — A B C to A B D {Art. 
499, obtuse angle) — then each of the two angles is called a 
right angle. 

327. — Definition. — The circumference of every circle 
is supposed to be divided into 360 equal parts, called 
degrees ; hence a semicircle contains 180 degrees, a quad- 
rant 90, etc. 

E 




328. — Definition, — The measure of an angle is the num- 
ber of degrees contained between its two sides, using the 
angular point as a centre upon which to describe the arc. 
Thus the arc C E {Fig. 250) is the measure of the angle 
CB E, EAoi the angle E B A, and A D oi the angle A B D. 

329. — Corollary. — As the two angles at B {Fig. 249) are 
right angles, and as the semicircle, CAD, contains 180 de- 
grees {Art. 327), the measure of two right angles, therefore, is 



RIGHT ANGLES AND OBLIQUE ANGLES. 349 

180 degrees; of one right angle, 90 degrees; of half a right 
angle, 45 ; of one third of a right angle, 30, etc. 

330. — Definition. — In measuring an angle {Art. 328), no 
regard is to be had to the length of its sides, but only to the 
degree of their inclination. Hence eqiial angles are such as 
have the same degree of inclination, without regard to the 
length of their sides. 

331- — Axiom.— If two straight lines parallel to one 
another, a.s A B and CD {Fig. 251), stand upon another 
straight line, as E F, the angles A B F and CDF are equal, 
and the angle A B F is equal to the angle CDF. 

332. — Definition. — If a straight line, as A B {Fig. 250), 
stand obliquely upon another straight line, as CD, then one 

A c 

/ 



E B D P 

Fig. 251. 

of the angles, sls A B C, is called a7i obtuse angle, and the 
other, as A B Dy an acute angle. 

333.— Axiom. — The two angles^ B DaxvdiA B C {Fig. 2^0) 
are together equal to two right angles {Arts. 326, 329) ; also, 
the three angles A B D, E B A, and CBE are together 
equal to two right angles. 

334. — Corollary. — Hence all the angles that can be 
made upon one side of a line, meeting in a point in that 
line, are together equal to two right angles. 

336, — Corollary. — Hence all the angles that can be made 
on both sides of a line, at a point in that line, or all the 
angles that can be made about a point, are together equal to 
four right angles. 



350 



GEOMETRY. 



336. — Propo§ition. — If to each of two equal angles a 
third angle be added, their sums will be equal. Let ABC 
and D E F {Fig. 252) be equal angles, and the angle I J K t\\Q 
one to be added. Make the angles G B A and H E D each 
equal to the given angle I J K\ then the angle G B C will be 
equal to the angle H E F\ for if ABC and D E F hQ angles 




of 90 degrees, and IJK 30, then the angles GBC and 
H EF will be each equal to 90 and 30 added, viz., 120 
degrees. 

337- — Proposition. — Triangles that have two of their 
sides and the angle contained between them respectively 
equal, have also their third sides and the two remaining 




angles equal ; and consequently one triangle will every way 
equal the other. Let ABC {Fig. 253) and D E F he two 
given triangles, having the angle at A equal to the angle at 
D, the side A B equal to the side D E, and the side A C 
equal to the side D F\ then the third side of one, B C, is equal 
to the third side of the other, E F\ the angle at B is equal to 
the angle at E. and the angle at C is equal to the angle at 



EQUAL TRIANGLES IN PARALLELOGRAMS. 



35 



F. For if one triangle be applied to the other, the three 
points B, A, C, coinciding with the three points E, D^ F, the 
line ^ 6^ must coincide with the line EF] the angle at B 
with the angle at E\ the angle at C with the angle at F\ 
and the triangle B A C hQ every way equal to the triangle 
EDF. 

338. — Propo§itioii. — The two angles at the base of an 
isosceles triangle are equal. Let ABC {Fig. 254) be an 




isosceles triangle, of which the sides, A B and A C, are equal. 
Bisect the angle {Art. 506) BAChy the line A D. Then, the 
line B A being equal to the line A C, the line A D oi the 
triangle E being equal to the line A D oi the triangle F 
(being common to each), the angle BAD being equal to the 
angle DA Cy — the line B D must, according to Art. 337, be 



C D D C 

Fig. 255. 

equal to the line D C, and the angle at B must be equal to 
the angle at C. 

339. — Proposition. — A diagonal crossing a parallelogram 
divides it into two equal triangles. Let CD EF {Fig. 255) 
be a given parallelogram, and C F 7i line crossing it diag- 
onally. Then, as ^ 6" is equal to F D, and E F io C D, the 
angle at E to the angle at D, the triangle A must, according 
to Art. 337, be equal to the triangle B. 



352 



GEOMETRY. 



340 Proposition. — Let J K LM {Fig. 256) be a given 

parallelogram, and K L^ diagonal. At any distance between 
yATand LM draw N P parallel to J K\ through the point 
G, the intersection of the lines K L and N P, draw HI 
parallel to K M. In every parallelogram thus divided, the 
parallelogram A is equal to the parallelogram B. For, ac- 
cording to Art. 339, the triangle J K L \s equal to the tri- 
angle K L M, the triangle C to the triangle D, and E to F; 



H K 




this being the case, take D and F from the triangle K LM, 
and C diud E from the triangle jf K L, and what remains in 
one must be equal to what remains in the other ; therefore, 
the parallelogram A is equal to the parallelogram B. 



34(, — Proposition. — Parallelograms standing upon the 
same base and between the same parallels are equal. Let 
A BCD and EFCD (Fig. 257) be given parallelograms 




Fig. 257. 

standing upon the same base, C D^ and between the same 
parallels, A F and CD. Then A B and E F, being equal to 
CD, are equal to one another; BE being added to both 
A B and E F, A E equals BE] the line A C being equal to 
B D, and A E to B F, and the angle C A E being equal {Art. 
331) to the angle Z>j5/^, the triangle ^4 ^ C must be equal 
{Art. 337) to the triangle BED; these two triangles being 
equal, take the same amount, the triangle BEG, from each, 



TRIANGLE EQUAL TO QUADRANGLE. 



353 



and what remains in one, A B G C, must be equal to what 
remains in the other, E F D G ; these two quadrangles being 
equal, add the same amount, the triangle C G D, to each, and 
they must still be equal ; therefore, the parallelogram 
A B CD is equal to the parallelogram E F C D. 

342. — Corollary. — Hence, if a parallelogram and triangle 
stand upon the same base and between the same parallels, 




Fig. 258. 

the parallelogram will be equal to double the triangle. 
Thus, the parallelogram A D {Fig. 257) is double {Art, 339) 
the triangle C E D. 

343. — Proposition. — Let F G H D {Fig. 258) be a given 
quadrangle with the diagonal F D. From G draw G E 



IE 



/Q- 
Fig. 259. 

parallel to FD-, extend H D to E\ join i^and E\ then the 
triangle F E H \n\\\ be equal in area to the quadrangle 
FGHD. For since the triangles FDG and FDE stand 
upon the same base, F D, and between the same parallels,. 
FD and G E, they are therefore equal {Arts. 341, 342) ; and 
since the triangle C is common to both, the remaining, tri- 



54 



GEOMETRY. 



angles, A and B, are therefore equal ; then, B being equal to 
A, the triangle F E H is equal to the quadrangle F G H D. 

34-4-. — Proposition. — If two straight lines cut each other, 
as FG and H J {Fig. 259), the vertical, or opposite angles, 
A and C, are equal. Thus, FE, standing upon H J^ forms 
the angles B and C, which together amount {Art. 333) to two 
right angles ; in the same manner, the angles A and B form 
two right angles ; since the angles A and B are equal to B 
and C^ take the same amount, the angle B, from each pair, 
and what remains of one pair is equal to what remains of 
the other ; therefore, the angle A is equal to the angle C. 
The same can be proved of the opposite angles B and D. 




345. — Proposition. — The three angles of any triangle are 
equal to two right angles. Let ABC {Fig. 260) be a given 
triangle, with its sides extended to /% E and D, and the line 
(7 6^ drawn parallel t'o BE. As 6* 6^ is parallel to EB,\\\q 
angle at H is equal {Art. 331) to the angle at Z ; as the lines 
FC 2indi BE cut one another at A, the opposite angles at M 
and A^are equal {Art. 334) ; as the angle at A^ is equal {Art. 
331) to the angle at J, the angle at J is equal to the angle at 
M\ therefore, the three angles meeting at C are equal to the 
three angles of the triangle A B C \ and since the three angles 
at C are equal (y^r/. 333) to two right angles, the three angles 
of the triangle ABC must likewise be equal to two right 
angles. Any triangle can be subjected to the same proof. 

346- — Corollary. — Hence, if one angle of a triangle be a 
right angle, the other two angles amount to just one right 
angle. 



RIGHT ANGLE IN SEMICIRCLE. 355 

347._CoroIlary.— If one angle of a triangle be a 'right 
ano-le and the two remaining angles are equal to one another, 
these are each equal to half a right angle. 

348.— Corollary — If any two angles of a triangle amount 
to a right angle, the remaining one is a right angle. 

349. —Corollary.— If any two angles of a triangle are to- 
gether equal to the remaining angle, that remaining angle is 
a right angle. 

350.— Corollary.— If any two angles of a triangle are each 
equal to two thirds of a right angle, the remaining angle is 
also equal to tw^o thirds of a right angle. 

35 [, —Corollary. — Hence, the angles of an equilateral 
triangle are each equal to two thirds of a right angle. 




352. — Proposition. — If from the extremities of the di- 
ameter of a semicircle two straight lines be drawn to any 
point in the circumference, the angle formed by them at that 
point will be a right angle. Let ABC {Fig. 261) be a given 
semicircle; and ^ ^ and i5 (7 lines drawn from the extrem- 
ities of the diameter A C to the given point B ; the angle 
formed at that point by these lines is a right angle. Join 
the point B and the centre D ; the lines DA, D B, and D C, 
being radii of the same circle, are equal; the angle at A is, 
therefore, equal {Art. 338) to the angle at E ; also, the angle 
at C is, for the same reason, equal to the angle at F; the 
angle ABC, being equal to the angles at A and C taken to- 
gether, must, therefore {Art. 349), be a right angle. 

353. — Proposition. — The square on the hypothenuse of a 
right-angled triangle is equal to the squares on the two re- 



35<^ 



GEOMETRY. 



maining sides. Let ABC {Fig. 262) be a given right-angled 
triangle, having a square formed on each of its sides ; then 
the square ^^ is equal to the squares //(7and GB taken 
together. This can be proved by showing that the parallelo- 
gram B L is equal to the square GB ; and that the parallelo- 
gram CL is equal to the square HC. The angle CBB is a 
right angle, and the angle A B Fis aright angle ; add to each 
of these the angle ABC; then the angle FB C will evidently 
be equal {Art. 336) to the angle ABB; the triangle FB C 
and the square G B, being both upon the same base, FB, and 
between the same parallels, FB and G C, the square G B is 
equal {Art. ^^42) to twice the triangle FBC; the triangle 
A B D and the parallelogram B L, being both upon the same 




Fig. 262. 

base, B D, and between the same parallels, B D and A L, the 
parallelogram B L is equal to twice the triangle ABD; 
the triangles, FB C and ABD, being equal to one another 
{Art. 337), the square G B is equal to the parallelogram B L, 
either being equal to twice the triangle F B C ox A B D. The 
method of proving Z/^' equal to C L is exactly similar — thus 
proving the square ^ ^ equal to the squares // (7 and 6^ ^, 
taken together. 

This problem, which is the 47th of the First Book of 
Euclid, is said to have been demonstrated first by Pythago- 
ras. It is stated (but the story is of doubtful authority) 
that as a thank-offering for its discovery he sacrificed a hun- 
dred oxen to the gods. From this circumstance it is some- 
times called the Jiecatomb problem. It is of great value in 



DIAGONAL OF SQUARE FORMING OCTAGON. 



357 



the exact sciences, more especially in Mensuration and As- 
tronomy, in which many otherwise intricate calculations are 
by it made easy of solution. 

354. — Proposition. — In an equilateral octagon the semi- 
diagonal of a circumscribed square, having its sides coinci- 
dent with four of the sides of the octagon, equals the dis- 
tance along a side of the square from its corner to the more 
remote angle of the octagon occurring on that side of the 
square. Let Fig. 263 represent the square referred to ; in 
which O is the centre of each ; then A O equals A D. To 
prove this, it need only be shown that the triangle A O D \^ 
an isosceles triangle having its sides A O and A D equal. The 




Fig. 263. 

octagon being equilateral, it is also equiangular, therefore 
the angles B C O, E C O, A D O, etc., are all equal. Of the 
right-angled triangle FEC,FC and FE being equal, the 
two angles FE C ^nd EC E, :xrQ equal {Art. 338), and are 
therefore {Af't. 347) each equal to half a right angle. In like 
manner it may be shown that FA B and FB A are also each 
equal to half a right angle. And since FE C and FA B are 
equal angles, therefore the lines E C and A B are parallel 
{Art. 331,) and hence the angles ECO and A OD are equal. 
These being equal, and the angles ^(S' (9 ?ind A D O being 
equal by construction, as before shown, therefore the angles 
A O D and ADO are equal, and consequently the lines A 
and A D are equal. {Art. 338.) 



358 GEOMETRY. . 

355- — Proposition. — An angle at the circumference of a 
circle is measured by half the arc that subtends it ; that is, 
the angle ABC {Fig. 264) is equal to half the angle ADC. 
Through the centre D draw the diameter BE. The tri- 
angle A B D \s an isosceles triangle, A Z>and B D being ra- 
dii, and therefore equal ; hence, the two angles at F and G 
are equal {Art. 338), and the sum of these two angles is equal 
to the angle at H {Art. 345), and therefore one of them, G, is 
equal to the half of H. The angles at H and at G (or ABE) 
are both subtended by the arc A E. Now, since the angle 




Fig, 264, 

at H is measured by the arc A E, which subtends it, there- 
fore the half of the angle at H would be measured by the 
half of the arc A E ; and since G is equal to the half of //, 
therefore G or A B E is measured by the half of the arc A E. 
It maybe shown in like manner that the angle E B C is 
measured by half the arc E C, and hence it follows that the 
angle A B C is measured by half the arc, A C, that sub- 
tends it. 

356. — Proposition. — In a circle all the inscribed angles, 
A, B, and C {Fig. 265), which stand upon the same side of the 



EQUAL ANGLES IX CIRCLES. 



359 



chord i?^ are equal. For each angle is measured by half 
the arc D F E {Art. 355). Hence the angles are all equal. 

357- — Corollary. — Equal chords, in the same circle, sub- 
tend equal angles. 




Fig. 265. 

358- — Proposition. — The angle formed by a chord and 
tangent is equal to any inscribed angle in the opposite seg- 




ment of the circle ; that is, the angle D {Fig. 266) equals the 
angle A. Let H F be the chord, and E G the tangent ; draw 
the diameter JH\ then JHG is a right angle, also J F H \s 



36o 



GEOMETRY. 



a right angle. {Art. 352.) The angles A and B together equal 
a right angle {Art. 346) ; also the angles B and D together 
equal a right angle (equal to the angle JHG): therefore, the 
sum oi A and B equals the sum of B and D. From each of 
these two equals, taking the like quantity B, the remainders 
A and D are equal. Thus, it is proved for the angle at A ; 
it is also true for any other angle ; for, since all other in- 
scribed angles on that side of the chord line H F equal the 
angle A {Art. 356), therefore the angle formed by a chord 
and tangent equals any angle in the opposite segment of the 
circle. This being proved for the acute angle D, it is also 
true for the obtuse angle EH F\ for, from any point, K {Fig, 
267) in the arc H K F, draw lines to J, F and H ; now, if it can 




be proved that the angle £i7i^ equals the angle FKH, the 
entire proposition is proved, for the angle F K H equals any 
of all the inscribed angles that can be drawn on that side of 
the chord. {Art. 356.) To prove, then, that jE"///^ equals 
H KF\ the angle ^///^ equals the sum of the angles A and 
B\ also the angle 77 A" /^equals the sum of the angles C and 
D. The angles B and D, being inscribed angles on the same 
chord, J F, are equal. The angles C and A, being right angles 
{Art. 352), are likewise equal. Now, since A equals 6" and B 
equals 7>, therefore the sum of A and B equals the sum of C 
and D — or the angle ^777^ equals the angle H K F. 

359. — Proposition. — The areas of parallelograms of 
equal altitude are to each other as the bases of the parallelo- 



PARALLELOGRAMS PROPORTIONATE TO BASES. 



361 



grams. In Fig. 268 the areas of the rectangles A B C D and 
B E D F are to each other as the bases CD and D F. For, 
putting the two bases in form of a fraction and reducing this 
fraction to its lowest terms, then the numerator and denomi- 
nator of the reduced fraction will be the numbers of equal 
parts into which the two bases respectively may be divided. 
For example, let the two given bases be 12 and 9 feet respect- 
ively, then ^ = f , and this gives four parts for the larger 
base and three parts for the smaller one. So, in Fig. 268, 
divide the base CD into four equal parts, and the base D F 
into three equal parts ; then the length of any one of the 
parts in CD will equal the length of any one of the parts in 
D F. Now, parallel with A C, draw lines from each point of 
division to the line A F, These lines will evidently divide 
the whole figure into seven equal parts, four of them occupy- 



A 






B 




E 
















C 






1 


D 




• 



Fig. 26S. 



ing the area A B C D, and three of them occupying the area 
B E D F, Now it is evident that the areas of the two rect- 
angles are in proportion as the number of parts respectively 
into which the base-lines are divided, or that — 

A B C D : B E D F : : C D : D F. 

The areas in this particular case are as 4 to 3. But in gen- 
eral the proportion will be as the lengths of the bases. 
Thus the proposition is proved in regard to rectangles, but 
it has been shown (Art-. 341) that all parallelograms of equal 
base and altitude are equal. Therefore the proposition is 
proved in regard to parallelograms generally, including rect- 
angles. 

360. — Proposition. — Triangles of equal altitude are to 
each other as their bases. It has been shown {Arf. 359) that 



362 GEOMETRY. 

parallelograms of equal altitude are in proportion as their 
bases, and it has also been shown {ArL 342) that of a triangle 
and parallelogram, when of equal base and altitude, the 
parallelogram is equal to double the triangle. Therefore 
triangles of equal altitude are to each other as their bases. 

36(. — Propo§moii. — Homologous triangles have their 
corresponding sides in proportion. Let the line CD {Fig. 
269) be drawn parallel with A B. Then the angles E C D 
and E A B are equal {Art. 331), also the angles E D C and 
E B A are equal. Therefore the triangles ECD and E AB 
are homologous, or have their corresponding angles equal. 




For, join C to B^ and A to D, then the triangles A C D and 
^ (7 i9, standing on the same base, (T/^, and between the 
same parallels, CD and A B, are equal in area. To each 
of these equals join the common area C D E, and the sums 
A DE and BCE will be equal. The triangles CD E and 
A D E, having the same altitude, are to each other as 
their bases C E and A E {Art. 360), or — 

CDE : ADE : : CE : AE. 

Also the triangles CDE and BCE, having the same alti- 
tude, are to each other as their bases D E and BE, or — 

CDE : BCE : : DE : BE. 



CHORDS GIVING EQUAL RECTANGLES. 



363 



And, since the triangles A D E and B C E are equal, as before 
shown, therefore, substituting in the last proportion A D E 
for B C E, we have — 

CDE : ADE : : DE : BE. 

The first two factors here being identical with the first two 
in the first proportion above, we have, comparing the two 
proportions — 

CE : AE : : DE : BE; 

or, we have the corresponding sides of one triangle, CB E, 
in proportion to the corresponding sides of the other, /^ BE. 




Fig. 270. 

362. — Proposition. — Two chords, ^/^ and CD {Eig. 270), 
intersecting, the parallelogram or rectangle formed by the 
two parts of one is equal to the rectangle formed by the 
two parts of the other. That is, the product oi C G multi- 
plied by G D is equal to the product oi E G multiplied by 
G F. The triangle A is similar to the triangle B, because it 
has corresponding angles. The angle H equals the angle G 
{Art. 344) ; the angle at J equals the angle at K, because 
they stand upon the same chord, D F {Art. 356) ; for the same 



364 GEOMETRY. 

reason the angle M equals the angle Z, for each stands upon 
the same chord, E C. Therefore, the triangle A having the 
same angles as the triangle B^ the length of the sides of one 
are in like proportion as the length of the sides in the other 
{Art. 361). So— 

CG : EG : '. GF \ G D, 

Hence, as the product of the means equals the product of 
the extremes {Art, 373), E G multiplied by G F \^ equal to 
C G multiplied by G D, 

363. — Proposition. — If the sides of a quadrangle are 
bisected, and lines drawn joining the points of bisection in 
the adjacent sides, these lines will form a parallelogram. 




Draw the diagonals A B and CD {Fig. 271). It will here be 
perceived that the two triangles A E O and ACD are homol- 
ogous, having like angles and proportionate sides. Two of 
the sides of one triangle lie coincident with the two corres- 
ponding sides of the other triangle, therefore the contained 
angles between these sides in each triangle are identical. 
By construction, these corresponding sides are proportion- 
ate ; A C being equal to twice A E, and A D being equal to 
twice A O ; therefore the remaining sides are proportionate, 
CD being equal to twice E O, hence the remaining corres- 
ponding angles are equal. Since, then, the angles A E O 
and ACD are equal, therefore the line E is parallel with 



PARALLELOGRAM IN QUADRANGLE. 365 

the diagonal CD — so, likewise, the line^^A^is parallel to the 
same diagonal, CD. If, therefore, these two lines, EO and 
M N, are parallel to the same line, CD, they must be parallel 
to each other. In the same manner the lines ON and EM 
are proved parallel to the diagonal A B, and to each other ; 
therefore the inscribed figure ME ON is 3, parallelogram. 
It may be remarked, also, that the parallelogram so formed 
will contain just one half the area of the circumscribing 
quadrangle. 



SECTION VII.— RATIO, OR PROPORTION. 

364-. — mercliaiKlise. — A carpenter buys 9 pounds of nails 
for 45 cents. He afterwards buys 87 pounds at the same 
rate. How much did he pay for them ? 

An answer to this question is readily found by multiply- 
ing the ^y pounds by 45 cents, the price of the 9 pounds, 
and dividing the product, 391 5, by 9; the quotient, 435 cents, 
is the answer to the question. 

365. — Tlie "Rule of Three." — The process by which 
this problem is solved is known as the Rule of Three, or 
Proportion. 

In cases of this kind there are three quantities given, to 
find a fourth. Previous to working the question it is usual 
to make a statement, placing the three given quantities in 
such order that the quantity which is of like kind with the 
answer shall occupy the second place ; the quantity upon 
which this depends for its value is put in the first place, and 
the remaining quantity, which is of like kind with that in the 
first place, is assigned to the third place. 

When thus arranged, the second and third quantities are 
multiplied together and the product is divided by the first 
quantity ; the quotient, the answer to the question, is a 
fourth quantity. These four quantities are related to each 
other in this manner, namely : the first is in proportion to 
the second as the third is to the fourth ; or, taking the 
quantities of the given example, and putting them in a for- 
mal statement with the customary marks between them, we 
have — 

9 : 45 :: 87 : 435, 

which is read: 9 is to 45 as 87 is to 435 ; or, 9 is in propor- 
tion to 45 as 87 is to 435 ; or, 9 bears the same relation to 45 
as 87 does to 435. 



EQUALITY OF RATIOS. 367 

366. — Couples: Antecedent, Consequent. — These four 
quantities are termed Proportionals, and may be divided into 
two couples ; the first and second quantities forming one 
couple, and the third and fourth the other couple. Of each 
couple the first quantity is termed the antecedent, and the 
last the consequent. Thus 9 is an antecedent and 45 its con- 
sequent ; so, also, Zj is an antecedent and 435 its consequent. 

367. — Equal Couples: an Equation. — These four quan- 
tities may be put in form thus : 

9 87 

Each couple is here stated as a fraction : each has its ante- 
cedent beneath its consequent, and the two couples are 
separated by a sign, two short parallel lines, signifying 
equality. This is an equation, and is read thus : 45 divided 
by 9 is equal to 435 divided by 87 ; or, as ordinary fractions : 
45 ninths are equal to 435 eighty-sevenths. 

368. — Equality of Ratios. — Each couple is also termed a 
Ratio, and the two the Equality of Ratios. Thus the ratio 

— is equal to the ratio -~^-. If the division indicated in 
9 87 

these two ratios be actually performed, the equality between 
the two will at once be apparent, for the quotient in each 
case is 5. The resolution of each couple into its simplest 
form by actual division is shown thus : 

435 _ , 
87 -5- 

These are read : 45 divided by 9 equals 5 ; and 435 divided 
by 87 equals 5. 

369. — Equals Iflultiplied by Equals Give Equals. — If two 

equal quantities be each multiplied by a given quantity, the 



368 RATIO, OR PROPORTION. 

two products will be equal. For example, the fractions f 
and |- are each equal to i, and are therefore equal to each 
other. If these two equal quantities be each multiplied by 
any given number, say, for example, by 4, we shall have 4 
times f equals |, and 4 times f equals -^2. ; these products, f 
and -V^- are each equal to 2, and therefore equal to each 
other. 

370. — Multiplying^ an Equation. — The quantity on each 
side of the sign = is called a meinber of the equation. If 
each member be multiplied by the same quantity, the 
equality of the two members is not thereby disturbed {Art. 
369) ; therefore, if the two members of the equation 

— = ^^ {Art. 367) be each multiplied by 87, or be modified 

thus : 

45x87 ^ 435 X 87 
9 87 ' 

in which x, the sign for multiplication, indicates that the 
quantities between which it is placed are to be multiplied 
together ; this ddition to each member of the equation does 
not destroy the equality ; the members are still equal, 
though considerably enlarged. The equality may be easily 
tested by performing the operations indicated in the equa- 
tion. For example : for the first member, we have 45 times 
87 equals 3915, and this divided by 9 equals 435. Again, for 
the second member we have 435 times 87 equals 37845, and 
this divided by 87 equals 435, the same result as that for the 
first member. Thus the multiplication has not interfered 
with the equality of the members. 

371. — Multiplying: and I>iTi«Sing: one Member of an 
Equation : Caneelling. — If a quantity be multiplied by a 
given number, and the product be divided by the same 
given number, the quotient will equal the original quantity. 
For example : if 8 be multiplied by 3, the product w^ill be 24 ; 
then if this product be divided by 3, the quotient w^ill be 8, 
the original quantity. Thus the value of a quantity is not 



TRANSFERRING FACTORS IN EQUATIONS. 369 

changed by multiplying it by a number, provided it be also 
divided by the same number. 

From this, also, we learn that the value of a quantity 
which is required to be multiplied and divided by the same 
number will not be changed if the multiplication and divis- 
ion be both omitted ; one cancels the other. Therefore the 
number 87, appearing in the second member of the equation 
in the last article both as a multiplier and a divisor, may be 
omitted without destroying the equality of the two mem- 
bers. The equation thus treated will be reduced to — 

45 X87 

^- = 435. 

This expression is read : the product of 45 times 87 divided 
by 9 equals 435. It will be observed that we have here the 
four terms of the problem in Art. 365, three of them in the 
first member, and the fourth, the answer to the problem, in 
the second member. 

372. — Transferring a Factor. — Each of the four quan- 
tities in the aforesaid equation is termed 2i factor. Compar- 
ing the equation of the last article with that of Art. 43, it 
appears that the two are alike excepting that the factor 87 
has been transferred from one member of the equation to 
the other, and that, Avhereas it was before a divisor, it has 
now become a multiplier. From this we learn that a factor 
may be transferred from one member of an equation to the 
other, provided that in the transfer its relative position to 
the horizontal line above or below it be also changed ; that 
is, if, before the transfer, it be below the line, it must be put 
above the line in the other member; or, if above the line, it 
must be put below, in the other member. For example : in 
the equation of the last article let the factor 9 be removed 
to the second member of the equation. It stands as a divi- 
sor in the first member ; therefore, by the rule, it must appear 
as a multipher after the transfer ; or — 

45 X 87 = 9 X 435; 



370 RATIO, OR PROPORTION. 

which is read, 45 times 87 equals 9 times 435. By actually 
performing the operations here indicated, we find that each 
member gives the same product, 3915; thus proving that 
the equality of the two members was not interfered with 
by the transfer. 

373. — Eqwality of Products: Mcaais and Extremes. — In 

Art. 366, the four factors are put in the usual form of four 

proportionals. A comparison of these with the four factors 

as they appear in the equation in the last article, shows that 

the first member contains the second and third of the four 

proportionals, and the second member contains the first and 

the fourth ; or, the first contains what are termed the 

means, and the second, the extremes. From this we learn 

that in any set of four proportionals, the product of the 

means equals the product of the extremes. As for example, 

3 6 

-:=:ii; SO, also, - = i-J, an equality of ratios: hence the 

four factors, 2, 3, 4, 6, are four proportionals, and may be 
put thus : 

Extreme, mean, mean, extreme. 
2:3:14:6 

and, as above stated, the product of the means (3x4 — ) 12, 
equals the product of the extremes (2x6=) 12. 

374 Ilomoloiii^ous Triangles Proportionate. — The 

discussion of the subject of Ratios has thus far been con- 
fined to its relations with the mercantile problem of Art. 
364. The rules of proportion or the equality of ratios 
apply equally to questions other than those of a mercantile 
character. They apply alike to all questions in which quan- 
tities of any kind are comparable. For example, in geome- 
try, lines, surfaces, and solids bear a certain fixed relation to 
one another, and are, therefore, fit subjects for the rules of 
proportion. It is shown, in Art. 361, that the correspond- 
ing sides of homologous triangles are in proportion to one 
another. Hence, when, of two similar triangles, two corres- 
ponding sides and one other side are given, then by the 
equality of ratios the side corresponding to this other side 



RATIOS APPLIED TO TRIANGLES. 37 1 

may be computed. For example : in two triangles, such as 
BCD and EAB (Fig. 269), having- their corresponding 
angles equal, let the side E C, in the triangle BCD, equal 12 
feet, and the corresponding side B A, in the triangle B A B, 
equal 16 feet, and the side B D, of triangle E CD, equal 14 
feet. Now, having these three sides given, how can we find 
the fourth ? Putting them in proportion, we have, as in 
Art. 361 — 

CE : AE : : BE : BE; 

and, substituting for the known sides, their dimensions, we 
have — 

12 : 16 : : 14 : ^ii ; 

and, by zir/. 373 — 

I2x B E = 16 X 14, 

Dividing each member by 12, gives — 

12 

Performing the multiplication and division indicated, we 
have — 

BB = ^^= i8|. 
12 

Thus we have the fourth side equal to i8f feet. 

375. — The Steelyard. — An example of tour proportion- 
als may also be found in the relation existing between the 
arms of a lever and the weights suspended at their ends. A 
familiar example of a lever is seen in the common steelyard 
used by merchants in weighing goods. This is a bar, A B, 
of steel, arranged as in Big. 272, with hooks and links, and a 
suspended platform to carry R, the article to be weighed ; 
and with a weight P, suspended by a link at B, from the bar 
A B, along which the weight P is movable. 

The entire load is sustained by links attached to the ful- 
crum, or point of suspension C. The apparatus is in equi- 
librium without R and P. In weighing any article, R, the 



3/2 RATIO, OR PROPORTION. 

weight P is moved along- the bar B C until the weight just 
balances the load, or until the bar A B will remain in a hori- 
zontal position. If the weight P be too far from the fulcrum 
C the end of the bar B will fall, but if it be too near it will 
rise. 

376, — The liCver Exemplified toy the Steelyard. — To 

exemplify the principle of the lever, let the bar A B {Fig. 
272) be balanced accurately with the scale platform, but 
without the weights R and P. Then, placing the article R 
upon the platform, move the weight P along the beam until 
there is an equilibrium. Suppose the distances A C and B C 
are found to be 2 and 40 inches respectively, and suppose 




Fig, 272, 

the weight P to equal 5 pounds, what at this pomt will be 
the weight of i^ ? By trial we shall find that R = 100 pounds. 
Again, if a portion of R be removed, then the weight P 
would have to be moved along the bar B C to produce an 
equilibrium ; suppose it be moved until its distance from C 
be found to be 20 inches, then the weight of R would be 
found to be 50 pounds, or — 

7? = 50 pounds. 

Again, suppose a part of the weight taken from R be re- 
stored, and the weight P, on being moved to a point re- 
quired for equilibrium, be found to measure 30 inches from 
C, then we shall find that — 

R— 'j^ pounds. 



RATIOS OF THE LEVER. 373 

Thus when — 

^(7 = 40, ;?= 100; or, —-=2.5; 

40 

BC=zo, R = 7S\ or, ^=2.5; 

BC^2o, R = So; or, 1^ =2.5; 

showing an equality of ratios ; or, in general, ^ 67 is in pro- 
portion to R, or — 

BC : R. 

If, instead of moving P along B C, its position be permanent, 
and the weight P be reduced as needed to produce equilib- 
rium with the various articles, R, which in turn may be 
put upon the scale ; then we shall find that if when the 
weight P equals 5 pounds the article R equals 100, and there 
is an equilibrium, then \vhen — 

p_- — ^^—.^,^^ j^ ^^jjl equal — x 100 = 90; 

8 8 

P = — x5 = 4, 7? will equal — x 100 = 80; 
10 ^ ^10 

7 7 

P— — X 5 = 3 . 5, R will equal — x 100 =: 70 ; 

and so on for other proportions^ and in every case we shall 
have the ratio -p equal 20, thus— 



R 
P 


= 


90 

4-5' 


= 20- 


R 
P 


= 


80 
4 


: 20; 


R 
P 


= 


70 
3-5" 


= 20. 



374 RATIO, OR PROPORTION. 

Thus we have an equahty of ratios in comparing the 
weights. 

Again, if the weight P and the article R be permanent in 
weight, and the distances A Cy B C he made to vary, then if 
there be an equiUbrium when A C is 2 and B C is 40, we 
shall find that when — 

8 8 

AC— — X 2 = 1-6; B C will equal — x 40 = 32 ^ 

A C = — x2=i-2; B C will equal — x 40 = 24 ; 
10 ^10 

AC— ~ x2==o-8; BC will equal — x 40 =16 ; 
10 ^ 10 ^ 

and so on for other proportions, and in every case we shall 

have the ratio -j-^r = 20; thus — 



BC _ 

A C 


32 
1.6 


1= 20; 


BC 
AC 


24 

1-2 


= 20; 


B C 


16 


= 20: 



AC~Q'% 

producing thus an equality of ratios in comparing the arms 
of the lever. From these experiments we have found, in 
comparing the article weighed with an arm of the lever, the 
constant ratio B C '. R, and when comparing the weights 
we have found the constant ratio P : R. Again, in com- 
paring the arms of the lever, w^e find the constant ratio 
A C : B C. Putting two of these couples in proportion, we 
have — 

A C : B C : : P : R. 

Hence {A?'t. 373) we have — 

ACxR = BCxP. 



PRINCIPLE OF THE LEVER DEMONSTRATED. 

Dividing both members by A C, we have — 

BCx P 



375 



R 



A C ' 



In a steelyard the short arm, A C, and the weight, or poise, 
P, are unvarying ; therefore we have — 



R 



BCx 



P 

AT' 



or, when 



A C 



is constant, we have- 



R : B C. 



Zn , — The Licver Principle Demoiistrated. — 1 he rela- 
tion between the weights and their arms of leverage may be 
demonstrated as follows : * 



M 



" G>f? 



— © 



Fig, 273. 

Let A B G H, Fig. 273, represent a beam of homogeneous 
material, of equal sectional area throughout, and suspended 
upon an axle or pin at Cy its centre. This beam is evidently 
in a state of equilibrium. Of the part of the beam A D G K, 
let E be the centre of gravity ; and of the remaining part, 
D B K H, let F be the centre of gravity. 

If the weight of the material in A B^ G Khe concentrated 
at F, its centre of gravity, and the weight of the material in 



* The principle upon which this demonstration is based may be found in an 
article written by the author and published in \}ciQ Matheinatical MontJily, Cam- 
bridge, U. S., for 1858, p. 77. 



376 RATIO, OR PROPORTION. 

DBKH be concentrated in F, its centre of gravity, the 
state of equilibrium will not be interfered with. Therefore 
let the ball R be equal in weight to the part A D G K, and 
the ball P equal to the weight of the part D B KH \ and let 
these two balls be connected by the rod E F. Then these 
two balls and rod, supported at C^ will evidently be in a 
state of equilibrium (the rod E F being supposed to be with- 
out weight). 

NoAv, it is proposed to show that i^ is to P as C F is to 
C E. This can be proved; for, since R equals the area 
AD G K and P equals the area DBKH, therefore R is in 
proportion to A D, ?iS P is to D B {Art. 359) ; or, taking the 
halves of these lines, R is in proportion to A y as P is to 
LB. 

Also, y L equals half the length of the beam ; for J D is 
the half of A D, and D L \s the half oi DB\ thus these two 
parts {JD + DL) equal the half of the two parts (AD-\-D B)\ 
or, J L equals the half of A B ; or, we have — 

•^ 2 ' 2 

Adding these two equations together, we have — 



Now, JD-r DL^ 7L, and AD + DB = AB', therefore, 

Thus we have A M — J L. From each of these equals 
take J M, common to both, then the remainders, A J and 
ML, will be equal ; therefore, AJ—CF. 

We have also MB z=z J L. From each of these equals 
take ML, common to both, and the remainders, JM and 
L B, will be equal ; therefore, LB = E C. As Avas above 
shown — 

R '. AJ '.'. P '. LB. 



TO FIND A FOURTH PROPORTIONAL. 377 

Substituting for A y and LB their values, as just found, 
we have — 

R : CF :: P : EC; 

from which we have {Art. 373) — 

FxCF=:Rx£C 

Thus it is demonstrated that the product of one weight into 
its arm of leverage, is equal to the product of the other 
weight into its arm of leverage : a proposition which is 
known as the law of the lever. 

378. — Any One of Four Proportionals may be Found. 

— Any three of four proportionals being given^ the fourth 
may be found ; for either one of the four factors may be 
made to stand alone ; thus, taking the equation of the last 
article, if we divide both members by CF {Art, 371), we 
have — 

Px CF _ RxEC 
CF ~ CF ' 

In the first member C F, in both numerator and denominator, 
cancel each other {Art. 371), therefore — 

„ RxEC 











CF ' 


so 


likewise we 


may 


obtain— 
R^ 

CF^ 

EC^ 


Px CF 
EC 

RxEC 
- P 

px CF 

- rj • 



R 



w^^mm^mm^m^mmi^m^^m^m^mm 



SECTION VIII.— FRACTIONS. 

379. — A Fraction ©efined. — As a fracture is a break or 
division into parts, so a fraction is literally a piece broken off; 
a part of the whole. 

The figures which are generally used to express a frac- 
tion show what portion of the whole, or of an integer, the 
fraction is : for example, let the line A B, {Fig. 274), be divided 
into five equal parts, then the line A C, containing three of 
those parts, will be three fifths of the whole line A B, and 

may be expressed by the figures 3 and 5, placed thus, -, 

which is known as a fraction and is read, three fifths. The 
number 5 below the line denotes the number of parts into 
which an integer or unit, A B, is supposed to be divided ; it 



1 I 



DEC B 

Fig. 274. 



is therefore called the denoviinator, and expresses th3 denom- 
ination or kind, whether fifths, sixths, ninths, or any number, 
into which a unit is supposed to be divided. The number 
3 above the line, denoting the number of parts contained in 
the fraction, is termed the imvierator, and expresses the 
number of parts taken, as 2, 3, 4, or any other number. 

380. — Crrapliieal llcpresciitatioii of Fractions : Effect 
of MultipBication. — In Fig. 275, let the line A B h^ di- 
vided into three equal parts ; the line CD into six equal 
parts; the line EF into nine equal parts; the line 6^ //into 
twelve equal parts, and the line J Kx^X-O fifteen equal parts. 
The lines AB, CD, EF, GH, and J K, being all of equal 
length. 



FRACTIONS ILLUSTRATED. 



379 



Then the parts of these hnes, A L, CM, EN, etc., may 

be expressed respectively by the fractions-,^,-, - and ---. 
^ ^ •" -^ 3 6 9 12 15 

In each case the figure below the line, as, 3, 6, 9, 12, or 15, 
expresses the number of parts into which the whole is di- 
vided, and the figure above the line, as 1,2, 3, 4, or 5, the 



I I 



I I I 



Fig. 275. 



number of the parts taken ; and, as the lines A L, CM, EN, 
etc., are all equal to each other, therefore these fractions are 
all equal to each other. If the numerator and denominator 
of the first fraction be each multiplied by 2, the products 
will equal the numerator and denominator of the second 
fraction ; thus — 



so, also, 



I 


X 


2 


nz 


2 


3 


X 


-7 


= 


6 


I 


X 


3 


= 


3. 


3 


X 


3 


:=. 


9 


I 


X 


4 


— 


4 


3 


X 


4 


— 


12 


I 


X 


5 


— 


5 


3 


X 


5 


= 


15 



and 
and 



Thus it is shown that when the numerator and denomi- 
nator of a fraction are each multiplied by the same factor, 
the product forms a new fraction which is of equal value 
with the original. 

In like manner we have, -, — , --, --, etc., each equal to 

8 12 16 20 

one fourth ; and which may be found by multiplying the 

numerator and denominator of - successively by 2, 3, 4, 5, etc. 

4 



38o FRACTIONS. 

381. — Form of Fraction Chang^cd toy Division. — By an 

operation the reverse of that in the last article, we may re- 
duce several equal fractions to one of equal value. Thus, if 
in each we divide the numerator and denominator by the 
same number, we reduce it to a fraction of equal value, but 
with smaller factors. 

For example, taking the fractions of the last article, f , -f, 
tV» tV> ^^t each be divided by a number which will divide 
both numerator and denominator without a remainder.* 

Thus, 



2 -^2 = I 

6-2 = 3' 


3-f-3 = I 
9-3 = 3 


4^4= I 


5-^5 = 1 


I2-^4 = 3'' 


15-^5 = 3 



As these fractions are shown {Art. 380) to be equal, and 
as the operation of dividing each factor by a common num- 
ber produces quotients which in each case form the same 
fraction, i, we therefore conclude that the numerator and 
denominator of a fraction may be divided by a common 
number without changing the value of the fraction. 

382. — Improper Fractions, — The fractions f , ^, f, etc., 
all fractions which have the numerator larger than the de- 
nominator are termed improper fractions. They are not im- 
proper arithmetically, but they are so named because it is an 
improper use of language to call that depart which is greater 
than the whole. 

As expressions of this kind, however, are subject to the 
same rules as those which are fractions proper, it is custom- 
ary to include them all under the technical X.^r\x\oi fractions. 
Expressions like these — all expressions in which one number 
is separated by a horizontal line from another number below 
it, or one set of numbers is thus separated from another set 
below it — may be called fractions, and are always to be un- 
derstood as . indicating division, or that the quantity above 
the line is to be divided by the quantity below the line. 

* Division is indicated by this sign -j-, which is read "divided by." 



IMPROPER FRACTIONS. 38 1 

^, Q 17 24 3 X 8 X 4 17X 82 ^ „ , . ^ . 

Thus,-. —^1 -~^> -y — ' etc., are all fractions, tech- 

3 5 3 2 X 12 125 ' ' 

nically, although each may be greater than unity. And it is 

understood in each case that the operation of division is re- 

1 T-i 9 24 ^3x8x4 ^,^, , ,. . 

quired. 1 hus, - = 3, ^ — — 8, = 4. When the divis- 

^ 3 3 2x12 ^ 

ion cannot be made without a remainder, then the fraction, 
by cutting the numerator into two, may be separated into two 
parts, one of which may be exactly divided, and the other 

1 7 
will be a fraction proper. Thus, the fraction — ■ is equal to 

J c 2 I c 

h— (for 15 + 2 — 17) ; and since — equals 3, therefore, 

17 15 2 '2 2 

— - =-— + -= 3+-= 3-. So, likewise, the fraction 

17x82 ^1394^ 1375^ 19 ^ ^^ _^ J^_^ jj J_9_ 
125 125 125 125 125 125- 

383. — Reduction of Mixed Niinitoers to Fractions. — By 

an operation the reverse oi that in the last article, a given 
mixed number (a whole number and fraction) can be put 
into the form of an improper fraction. 

This is done by multiplying the whole number by the de- 
nominator of the fraction, the product being the numerator 
of a fraction equal in value to the whole number ; the de- 
nominator of this fraction being the same as that of the given 
fraction. The numerator of this fraction being added to the 
numerator of the given fraction, the sum will be the numera- 
tor of the required improper fraction, the denominator of 
which is the same as that of the given fraction. For example, 
the required numerator for — 

2I is 2 X 3 + I = 7. So 2I = |. 
2j, is 2 X 4 + I = 9. So 2^ = f . 
3f, is 3 X 5 +2 = 17. So 3-1 = 'i. 

384-. — Division Indicated by the Factors put as a Frac- 

2 c 1 20 
lion. — Factors placed in the form of a fraction as -, -, -^^or 

^ 5 3 75 



382 FRACTIONS. 

indicate division {Art. 382) ; the denominator (the fac- 
tor below the line) being the divisor, and the numerator 
(the factor above the line) the dividend, while the value of 

the fraction is the quotient. Thus of the fraction, -^- = 20, 

41 is the divisor, 820 the dividend, and 20 the quotient. 
From this we learn that division may always be indicated 
by placing the factors in the form of a fraction, so that the 
divisor shall form the denominator and the dividend the nu- 
merator. 



385. — A<l<lition of Fractions liaviii§^ ILikc Deiioinina- 

I 2 

tors. — Let it be required to add the fractions — and — . By 

referring to Art. 379 we see that A D {Fig. 274), is one of the 
five parts into which the whole line A B is divided ; it is, 

therefore, -. We also see that D C contains two of the five 

5 

2 
parts ; it is, therefore, -. We also see that AD + DC^=AC, 

which contains three of the five parts, or A C =^ - oi A B. 

123 
We therefore conclude that — + — = —. In this operation it 

is seen that the denominator is not changed, and that the 
resultant fraction has for a numerator a number equal to the 
sum of the numerators of the fractions which were required 
to be added. 

By this it is shown that to add fractio7is we simply take 
the Sinn of the mimcrators for the nezu imnierator, making the 
de?iominator of the resultant fraction the same as that of the 
fractions to be added. For example: What is the sum of the 

fractions -, - and -? Here we have 1+3 + 4 = 8 for the 
9 .9 9 



numerator, therefore — 



1348 

- + ^ + i — -. 

9 9 9 9 



SUBTRACTION AND ADDITION OF FRACTIONS. 383 

386.— Subtraction of Fractions of Like Denominators.— 

Subtraction is the reverse of addition ; therefore, to sub- 
tract fractions a reverse operation is required to that had in 
the process of addition ; or simply to subtract instead ot 
adding. 

For example, if - be required to be subtracted from — 
5 5 

we have — 



5 5 5 

By reference to Fig. 274 an exemplification of this will be 

32 I 

seen where we have A C ^^ ~, A E =z —^ and E C ■= -, and 

we have — 

A C - A E =. E C\ 
3 _ 2 ^ I 

5 5 "5' 

We therefore have this rule for the substraction of frac- 
tions: Subtract the less from the greater numerator ; the remain- 
der IV ill be the nnmcrator of the required fraction. The denom- 
inator to be the same as that of the given fractions. 

387. — Hissimilar I>enominator§ B^iualized. — The rules 
just given for the addition and subtraction of fractions re- 
quire that the given fractions have like denominators. 
When the denominators are unlike it is required, before add- 
ing or substracting, that the fractions be modified so as to 
make the denominators equal. For example ; Let it be re- 

2 2 

quired to find the sum of - and --. Bv reference to Fio-. 

3 9- 

2 . 6 

275, we find that - on line A B is equal to- on line E F. 

3 9 

6 2 
These being equal, we may therefore substitute - for -. 

Then we have — 

6 2 _ 8 
9 ^ 9 "~ 9 



384 FRACTIONS. 

Now, it will be seen that the fraction - may be had by mul- 
tiplying both numerator and denominator of the given frac- 

2, , 2x3=6 

tion - by 3, for ^ ^ ;^ _ - ; 

and we have seen {Art. 380) that this operation does not 
change the value of the fraction. From this we learn that 
t/ie denominators may be made equal by multiplying the smaller 
defwminator and its numerator by any number wJiich will effect 
such a result. 



For example 



15 15"^ 15 15 



27 14 7 21 
and --+ — = — + — = — ; 

5 35 35 35 35 ' 

'^'''^ 4 12 "^ 16 - 16 + 16 +T6 - 16 - ' i6- 

In this example the second fraction is changed by multiply- 
ing by i^. 

388, — Reduction of Fractions to tlieir L-oivest Terms. — 

The process resorted to in the last article to equalize the 
denominators, is not always successful. What is needed for 
a common denominator is to find the smallest number 
which shall be divisible by each of the given denominators. 
Before seeking this number, let each given fraction be 
reduced to its lowest terms, by dividing each factor by a 

common number. For example : — may, by dividing by 5, 

I . . . 21 . 

be reduced to -, which is its equivalent. So, also, — , by di- 

3 2b 

3 
viding by 7, is reduced to -, its lowest terms. 

389. — Licast Common Denominator. — To find the least 
common denominator, place the several fractions in the order 
of their denominators, increasing toward the right. If the 
largest denominator be not divisible by each of the others, 
double it ; if the division cannot now be performed, treble 



LEAST COMMON DENOMINATOR. 385 

it, and so proceed until it is multiplied by some number 
which will make it divisible by each of the other denomina- 
tors. This number multiplied by the largest denominator zvill be 
the least common denominator. To raise the denominator of 
each fraction to this, divide the common denominator by the de- 
nominator of one of the fractions, the quotient will be the 
number by which that fraction is to be multiplied, both 
numerator and denominator, and so proceed with each frac- 
tion. For example : What is the sum of the fractions 

-, -, — , - ? One of these, — , may be reduced, by divid- 

2' 4' 12' 8 ' 12' -^ * -^ 

ing by 2, to -p. Therefore, the series is -, -, ^, -. On trial 

we find that 8, the largest denominator, is divisible by the 
first and by the second, but not by the third, therefore the 
largest denominator is to be doubled : 2x8=16. This is 
not yet divisible by the third ; therefore 3 x 8 = 24. This 
now is divisible by the third as well as by the first and the 
second ; 24 is therefore the least common denominator. 

Now dividing 24 by 2, the first denominator, the quotient 
12 is the factor by which the terms of the first fraction are 

1 X 12= 12 

to be raised, or, - " -^. For the second we have 

2 X 12 = 24 

24 -f- 4 = 6, and - ^ — . For the third we have 24 -j- 6 = 
^^ 4x6 = 24 ^ 

4, and ^ _ — ; and for the fourth, 24-^8 = 3, and 
0x4 — ^4 

7 X ■^ =: 2 I 

Thus the fractions in their reduced form are : 



x3 =24 



12 18 20 21 71 23 

24 24 24 24 "" 24 ~ 24' 



390. — L.ea$t Common Denominator Ag^ain. — When the 
denominators are not divisible by one another, then to ob- 
tain a common denominator, it is requisite to multiply to- 
gether all of the denominators which zvill not divide any of the 
other denominators. For example : What is the sum of the 

fractions -, -, -, and -? 



386 



FRACTIONS. 



In this case the first denominator will divide the last, but 
the others are prime to each other. Therefore, for the 
common denominator, multiply together all but the first ; 
or — 

5 x7x9 = 3i5 the common denominator ; 

and — 

315-^3 = 105, common factor for the first fraction ; 
315 -^ 5 =63, common factor for the second fraction ; 



315--7 = 45> 



common factor for the third 



315-^-9 = 35, common factor for the fourth. 
And, then — 



_i X 105 = 105 ^ ^ X 63 = 1^26 _3 X 45 = 1^5 _ 4 X 35 = 240 
3 X 105 = 315 ' 5 X 63 = 315 ' 7 X 45 = 315 ' 9 X 35 = 315 



105 126 
"31^5 "^ 3^15 



i_35 
315 



140 
3"^ 



506 _ 191 



391. — Fractions Multiplied Graphically. — Let A B C D 

{Fig. 276) be a rectangle of equal sides, or A B equal A C 
and each equal one foot. Then A B multiplied hy A C will 




Fig. 276. 



equal the area A B C D, or i x i = i square foot. Let the 
line B Fhe parallel with A B, and midway between A B and 
CB. Then AB x A £ equals half the area of AB CD, or 
J X I = J^. Again ; let G H be parallel with E C, and mid- 
way between E C and FD. Then EGy.EC = iy.i equals 
the area E G C H, which is equal to a quarter of the area 



MULTIPLICATION OF FRACTIONS. 



,87 



A B C D; or i X i = i; which is a quarter of the siiperhcial 

area. 

The product here obtained is less than either of the 
factors producing it. It must be remembered, however, 
that while the factors represent /hies, the product represents 
superficial area. The correctness of the result may be 
recognized by an inspection of the diagram. 

392. — Fractions Multiplied Graphically. — In i^^> 277 
let A B equal 8 feet and A C equal 5 feet ; then the rect- 

































F 


















































A 














a fl 



Fig. 277. 

angle A BCD contains 5 x 8 = 40 feet. The interior lines 
divide the space included within A B CD into 40 equal 

squares of one foot each. Let A E equal 3 feet or - of ^ C. 
Let A G equal 7 feet or | of A B. Then the rectangle 

1 1 ? T 

E F A G contains -- x (- = — , or twenty-one fortieths of the 
5 8 40 

whole area ABC D, Thus, while the factor fractions -; and - 

21 
represent lines, it is shown that the product fraction -— rep- 

■^ T 

resents surface. Thus — is a fraction, E FA G, of the whole 

40 

surface, C D A B. 

393.— Rule for Mutiplication of Fractions, and Exam- 
ple. — In the example given in the last article it will be ob- 



388 FRACTIONS. 

served that the product of the denominators of the two 
given fractions equals the area of the whole figure {A B CD), 
while the product of the numerators equals the area of the 
rectangle {E F A G), the sides of which are equal respec- 
tively to the given fractions. From this we obtain for the 
product of fractions this — 

Rule. — Mtdtiply togetJier the denominators for the new de- 
nominator, and tJie numerators for a new numerator. 

For example : what is the product of —and — ? Here 

w^e have 20x21=420 for the new denominator, and 
7 X 13 = 91 for the new numerator ; therefore the product 
of— 

2 1 20 420 ' 

or, of a rectangular area divided one way into 20 parts and 
the other way into 21 parts, thus containing 420 rectangles, 

13 7 

the product of the two fractions — and — is equal to 91 of 

these rectano:les, or of the whole. 

^ ' 420 

394. — Fractions Divided Graphically. — Division is the 

reverse of multiplication ; or, while multiplication requires 
the product of two given factors, division requires one of 
the factors when the other and the product are given. Or 
(referring to Fig. 277) in division we have the area of the 
rectangle, E FA G, and one side, E A, given, to find the 
other side, A G. 

Now it is required to find the number of times E A \s 
contained in E FA G. By inspection of the figure we per- 
ceive the answer to be, A G times ; for E A y. A G — E FA G, 

2 1 
the given area. Or, when E A EG is given as — and E A 

as -, we have as the given problem — 



40 



DIVISION OF FRACTIONS. 3S9 

Since division is the reverse of multiplication, instead of 
multiplying we divide the factors, and have — 

21 -^ ^ 7 



40-^ 5 « 

Thus, to divide one fraction by another, for the numerator of 
the required factor, divide the numerator of tJie product by the 
numerator of the given factor, and for the denominator of the 
required factor divide the denominator of the product by the 
denominator of the given factor. For example : 

10 2 5 

Divide 7- by -. Answer, -. 
63-9 7 

Divide ^^ by — . Answer, — . 

27 -^ 9 3 

395. — Rule for Divmoii of Fractions. — The rule just 
given does not work well when the factors are not commen- 

• • 5 2 
surable. For example, if it be required to divide - by - we 
^ 7-^9 

have by the above rule — 

_5_ 

5 -=- 2 _ 2 

7-9 ~ y * 

9 

Producing fractional numerators and denominators for the 
resulting fraction, which require modification in order to 
reach those composed only of whole numbers. If the nu- 
merators, 5 and 7, of this compound fraction be multiplied 
by 9 (the denominator of the denominator fraction), or the 
compound fraction by 9, we shall have — 

5 5x9 



X 



9 



7 7x9 



390 FRACTIONS. 

And, if these be again multiplied by 2 (the denominator of 
the numerator fraction), we shall have — 

5x9 5x9x2 

2 2 

X 2 = 



7x9 7x9x2 

9 9 

Like figures above and below in each fraction cancel each 
other {Art. 371), therefore, the result reduces to — 

5 X 9 
7x2' 

in which we find the factors of the two original fractions. 

In one fraction — we have the factors in position as given, 

2 
but in the other — they are inverted. The fraction in which 

the factors are inverted is the divisor. Hence, for division 
of fractions, we have this — 

Rule. — Invert the factors of the divisors and then, as in 
multiplication^ multiply the numerators together for the numera- 
tor of the required fraction, aiid the denominators for the de- 
nominator of the required fractiofi. 

f . 2 

Thus, as before, if - is required to be divided by -, we 

have — 

5_x9^45 

7 X 2 14* 

2"^ 7 

And, to divide — by — , we have — 
47 ^ 9 

23 X 9 _ 207 

"47 X 7 "" 329* 

As^ain, to divide -- by -, we have — 
^ 45 -^ 9 

_2 s X 9 ^ 225 ^ ^ ^ ^ 
45 X 8 ~ 360 ~ 40 ~ 8* 



CANCELLING IN ALGEBRA. 39I 



This last example has two factors, 9 and 45, one of which 

45 



25 
measures the other ; also, the first fraction — is not in its 



lowest terms ; when reduced it is — . The question, there- 
fore, may be stated thus : 

5 x9 _ 5 
9 X 8 ~ 8' 

for the two 9's cancel each other. 



SECTION IX.— ALGEBRA. 

396. — Algebra Dcflned. — It occurs sometimes that a 
student familiar only with computation' by numerals is 
needlessly puzzled, in approaching the subject of Algebra, 
to comprehend how it is possible to multiply letters together, 
or to divide them. To remove this difficulty, it may be suf- 
ficient for them to learn that their perplexity arises from a 
misunderstanding in supposing the letters themselves are 
ever multiplied or divided. It is true that in treatises on 
the subject it is usual to speak as though these operations 
were actually performed upon the letters. It is always un- 
derstood, however, that it is not the letters, but the qtian- 
tities represented by the letters, which are to be multiplied 
or divided. 

For example, in Art. 361 it is shown, in comparing similar 
sides of homologous triangles, that the bases of the two tri- 
angles are to each other as the corresponding sides, or, 
referring to Fig. 269, Ave have C E -. A E -. -. D E \ B E. 
Now, let the two bases C E and A E he represented respec- 
tively by a and d, and the two corresponding sides E) E and 
B E by c and d respectively ; or, for — 

CE : AE : : DE : BE, 
put — 

a \ b \ \ c \ d\ 

and, by Art. 373, we have — 

b X c ^ ax d^ 

which may be written — 

be =^ ad', 

for X, the sign for multiplication, is not needed between let- 
ters, as it is between numeral factors. The operation of 



APPLICATION OF ALGEBRA. 393 

multiplication is always understood when letters are placed 
side by side. 

Now, here we have an equation in which, as usually read, 
we have the product of b and c equal to the product of a 
and d. But the meaning is that the product of the quantities 
represented by b and c is equal to the product of the quan- 
tities represented by a and d, and that this equation is in- 
tended to represent the relation subsisting between the four 
proportionals, C E, A E, D E, and BE, o{ Eig. 269. In order 
to secure greater conciseness and clearness, the four small 
letters are substituted for the four pair of capital letters, 
which are used to indicate the lines of the figures referred to. 

397- — Example : Application. — It was shown in the last 
article that the four letters a, b, c, and d represent the cor- 
responding sides of the two triangles of Eig. 269, and that — 

b c ^ a d. 

Now, let each member of this equation be divided by a, then 
{Art. 371)— 

a 

If now the dimensions of the three sides represented by a, 
b, and c are known, and it is required to ascertain from these 
the length of the side represented by d, let the three given 
dimensions be severally substituted for the letters repre- 
senting them. For example, let a = 40 feet; b = ^2 feet, 
and c = 4S feet ; then — 

be 52 X 45 2340 
d = — = ^—^= -^^ = 58.5 feet. 
a 40 40 -^ -^ 

The quantities being here substituted for the letters ; we have 
but to perform the arithmetical processes indicated to obtain 
the arithmetical value of d. From this example it is seen 
that before any practical use can be made of an algebraical 
formula in computing dimensions, it is requisite to substitute 
numerals for the letters and actually perform arithmetically 
such operations as are only indicated by the letters. 



394 ALGEBRA. 

398, — Alg^ebra Useful in Constructing Rules. — In all 

problems to be solved there are certain conditions or quan- 
tities given, by means of which an unknown quantity is to 
be evolved. For example, in the problem in Art. 397, there 
were three certain lines given to find a fourth, based upon 
the condition that the four lines were four proportionals. 
Now, it has been found that the relation between quantities 
and the conditions of a question can better be stated by let- 
ters than by numerals ; and it is the office of algebra to 
present by letters a concise statement of a question, and by 
certain processes of comparison, substitution and elimina- 
tion, to condense the statement to its smallest compass, and 
at last to present it in a formula or rule, which exhibits the 
known quantities on one side as equal to the unknown on 
the other side. Here algebra ends, at the completion of the 
rule. To 2ise the rule is the office of arithmetic. For, in 
using the rule, each quantity in numerals must be substi- 
tuted for the letter representing it, and the arithmetical 
processes indicated performed, as was done in Art. 397. 

399.— Algebraic Rules are General. — One advantage 
derived from algebra is that the rules made are general 
in their application. For example, the rule of Art. 397, 

— = d, is applicable to all cases of homologous triangles, 

however they may differ in size or shape from those given in 
Fig. 269 — and not only this, but it is also applicable in all 
cases where four quantities are in proportion so as to con- 
stitute four proportionals. For example, the case of the 
four proportionals constituting the arms of a lever and the 
weights attached {Arts. 375-378). For, taking the rela- 
tion as expressed in Art. 377 — 

PxCF= RxEC, 

we may substitute for C F the letter n, and for E C the letter 
in, then 7?t will represent the arm of the lever E C {Fig. 262), 
and n the arm of the lever F C. Then we have — 

Pn^Rm, 



SYMBOLS CHOSEN AT PLEASURE. 395 

and from this, dividing by 11 {A^rt. 372), we have — 

P=.R-) (no.) 

or, dividing by m, we have — 

R = ^'-, (.11.) 

which is a rule for computing the weight of R, when P and 
the two arms of leverage, 7n and n, are known. For example, 
let the weight represented by P be 1200 pounds, the length 
of the arm m be 4 feet, and that of n be 8 feet, then we have — 

Pn 1200x8 , 

R — — ■ = • = 2400 pounds. 

m 4 t r 

Pn 
This rule, R = — , is precisely like that in ArL 397 — 

— = d — in which three quantities are given to find a fourth, 
the four constituting a set of four proportionals. 

400. — Symbols Chosen at Pleasure, — The particular 
letter assigned to represent a particular quantity is a matter 
of no consequence. Any letter at will may be taken ; but 
w^hen taken, it must be firmly adhered to to represent that par- 
ticular quantity, throughout all the modifications which may 
be requisite in condensing the statement into which it enters 
into a formula for use. For example, the two rules named in 
Art. 399 are precisely alike — three quantities given to find a 
fourth — yet they are represented by different letters. In one, 
/? and /^ represent the two weights, and 7n and ?/ the arms of 
leverage at which they act ; while in the other the letters 
a, <^, <:, and <^ represent severally the four lines which constitute 
two similar sides of two homologous triangles. The two 
rules are alike in working, and they might have been con- 
stituted with the same letters. And instead of the letters 
chosen any others might have been taken, which con- 
venience or mere caprice might have dictated. In some 



39^ ALGEBRA. 

questions it is usual to put the first letters, as a, b, c, etc., to 
represent known quantities, and the last letters, as x, y, z, 
for the quantities sought. In works on the strength of 
materials it is customary to represent weights by capital 
letters, as P, R, U, W, etc., and lines or linear dimensions by 
the small letters, as b, d, /, for the breadth, depth, and length, 
respectively, of a beam. Any other letters may be put to 
represent these quantities, although the initial letter of the 
word serves to assist the memory in recognizing the partic- 
ular dimensions intended. 

401.^ — Arithmetical Proce§§e§ Indicated by Signs. — In 

algebra, the four processes of addition, subtraction, multi- 
plication, and division, are frequently required ; and when 
the required process cannot be actually performed upon the 
letters themselves, a certain method has been adopted by 
which the process is indicated. For example, in additon, 
when it is required to add a to by the two letters cannot be 
intermingled as numerals may be, and their sum presented ; 
but the process of addition is simply indicated by placing 
between the two letters this sign, +, which is called plus, 
meaning added to ; therefore, to add ^ to ^ we have — 

a^b, 

Avhich is read a plus b, or the sum of a and b. When the 
quantities represented b}^ a and b are substituted for them — 
and not till then — they can be condensed into one sura. 
For example, let a equal 4 and b equal 3, then for — 

a-^b 
we have — 

4+3; 

and we may at once write their sum 7, instead of 4+ 3. 

So, likewise, in the process of subtraction, one letter can- 
not be taken from another letter so as to show how much of 
this other letter there will be left as a remainder ; but the 
process of subtraction can be indicated by a sign, as this, — , 
which is called minus, less, meaning subtracted from. For 



ALGEBRAICAL SIGNS. 39/ 

example, let it be required to subtract b from a. To do 
this we have — 

a — b\ 

which is read a minus b, and when the values of a and b are 
substituted for them, we have, when a equals 4, and b 
equals 3 — 

a — b, 
or — 

4-3; 

and now, instead of 4 — 3, we may put the value of the two, 
which is unit}^, or i. 

The algebraic signs most frequently used are as follows : 

+ , //?^^, signifies addition, and that the two quantities be- 
tween which it stands are to be added together ; as 
a-^b, read a added to b. 

— , minus, signifies subtraction, or that of the two quantities 
between which it occurs, the latter is to be subtracted 
from the former ; 2iS> a — b, read a minus b. 

X, multiplied by, or the sign of multiplication. It denotes 
that the two quantities between which it occurs are to 
be multiplied together ; as <^ x ^, read a multiplied by b, 
or a times b. This sign is usually omitted between 
symbols or letters, and is then understood, as a b. This 
has the same meaning as a x b. It is never omitted 
between arithmetical numbers; as 9x5, read nine 
times five. 

-^, divided by, or the sign of division, and denotes that of the 
two quantities between which it occurs, the former is 
to be divided by the latter ; as a-^b, read a divided by 
b. Division is also represented thus : 

-, in the form of a fraction. This signifies that a is to be 
divided by b. When more than one symbol occurs 

above or below the line, or both, as , it denotes 

evi 

that the product of the symbols above the line is to be 
divided by the product of those below the line. 



39^ ALGEBRA. 

= , is equal to, or sign of equality, and denotes that the 
quantit}^ or quantities on its left are equal to those on 
its right ; as ^ — ^ = r, read a minus b is equal to c, or 
equals c \ or, 9 — 5 = 4, read nine minus five equals 
four. This sign, together with the symbols on each 
side of it, when spoken of as a whole, is called an 
equation. 

a" denotes a squared, or a multiplied by a, or the second 
power of a, and 

a^ denotes a cubed, or a multiplied by a and again multi- 
plied by a, or the third power of a. The small figure, 
2, 3, or 4, etc., is termed the index or exponent of the 
power. It indicates how many times the symbol is to 
be taken. Thus, a^ — a a, a^ — a a a, a" — a a a a. 
\/ is the radical sign, and denotes that the square root of the 

quantity following it is to be extracted, and 
4/ denotes that the cube root of the quantity following it is 
to be extracted. Thus, 1/9 = 3, and ^\/2j = 3. The 
extraction of roots is also denoted by a fractional in- 
dex or exponent, thus — 

rt:^ denotes the square root of a, 

a^ denotes the cube root of a, 

a^ denotes the cube root of the square of a, etc. 

402a — ExampSe in Addition and ISubtraction : Cancel- 
ling. — Let there be some question which requires a state- 
ment to represent it, like this — 

a + d =: c — b, 
which indicates that if the quantity represented by a be 
added to the quantity represented by d, the sum will be 
equal to the quantity represented by c, after there has been 
subtracted from it the quantity represented by b ; or, as it is 
usually read, a plus d equals c minus b; or the sum of a and 
d equals the difference between c and b. For illustration, 
take in place of these four letters, in the order they stand, 
the numerals 4, 2, 9, 3, and we shall have by substitution — 

a -{- d ^= c — b, 

4+2 = 9 — 3, or adding 

and subtracting — 6 — 6. 



TRANSFERRING SYMBOLS. 399 

If it be required to add to each member of the equation 
the quantity represented by b, this will not interfere with 
the equality of the members. For a^d are equal to f — d, 
and if to each of these two equals a common quantity be 
added, the sums must be equal ; therefore— 

a + d+b =c — b-^by 
or by numerals — 

4 + 2 + 3 = 9-3 + 3» 
or — 

9 = 9- 

It will be observed that the right hand member contains 
the quantity — b and + b. This shows that the quantity b is 
to be subtracted and then added. Now, if 3 be subtracted 
from 9, the remainder v/ill be 6, and then if 3 be added, the 
sum will be 9, the original quantity. Thus it is seen that 
whejz in the same member of an equation a symbol appears as a 
minus quantity and also as a plus quantity^ the two cancel each 
other, and may be omitted. Therefore, the expression — 

a + d + b ^=^ c — b + b 
becomes — 

a + d+ b —c. 

403. — Transferring a Symbol to the Opposite Meintoer. 

— In comparing, in the last article, the first equation with the 
last, it will be seen that the same symbols are contained in 
each, but differently arranged : that while in the first equa- 
tion b appears in the right hand member and with a minus 
or negative sign, in the last equation it appears in the left 
hand member and with a plus or positive sign. Thus it is 
seen that in the operation performed b has been made to 
pass from one member to the other, but in its passage it has 
been changed. A similar change may be made with another 
of the symbols. For example, from the last equation, let d 
be subtracted, or this process indicated, thus — 

a-\-d-\-b — d^=^c — d. 



400 ALGEBRA. 

The plus and minus d, in the left hand member cancel each 
other, therefore — 

a-\-b =^ c — d^ 
or, by numerals — 

4 + 3 = 9-2. 
Reducing — 

By this we learn that any quantity (connected by + or — ) 
may be passed from one member of the equation to the other, pro- 
vided the sign be changed. 

404. — Sig^ns of Symtools to be Changed wlien tliey are 
to toe Subtracted, — As an example in subtraction, let the 
quantities represented hy ^-b — a — /+ c, be taken from the 
quantities represented by +a+b — c—f This may be 
written — 

{+a + b — c —f) — {-{-b — a — /+ c), 

an expression showing that the quantities enclosed within the 
second pair of parentheses are to be subtracted from those 
mcluded within the first pair. Let the quantities represent- 
ed in the first pair of parentheses for convenience be repre- 
sented by y4 , or, a + b — c — / = A . Now, by the terms of the 
problem, we are required to subtract from A the quantities 
enclosed within the second pair of parentheses. To do this 
take first the positive quantity, b, and subtract it or indicate 
the subtraction, thus — 

A-b', 

we will then subtract the positive quantity c, or indicate the 
subtraction, thus — 

A-b-c. 

We have yet to subtract — a and — /, two negative quanti- 
ties. 

The method by which this can be accomplished may be 
discovered by considering the requirements of the problem. 
The plus quantities b and c, before being subtracted from A, 
were required to have the two negative quantities <3: and /de- 



THE OPERATION TESTED. 4OI 

ducted from them. It is evident, therefore, that in subtract- 
ing b and c, before this deduction was made, too much has 
been taken from A, and that the excess taken is equal to the 
sum of a and /. To correct the error, therefore, it is neces- 
sary to add just the amount of the excess, or to add the sum 
of a and/", or annex them by the plus sign, thus — 

A — b — c + a+f. 

To test the correctness of the operation as here performed, 
let numerals be substituted for the symbols ; l^t a-=2,b— 3, 
c =z i^ f— ^] then the given quantities to be subtracted, — 

(+b — a—f+c), 
become — 

(+3-2-1+1), 
which reduces to — 

(4 - 2i) = li. 

Thus the quantity to be substracted equals i^. Applying 
the numerals to the above expression — 

A — b+a-^f— c 
becomes — 

A — 3 + 2 +i— I = A — 4-^2^ = A — i^. 

A correct result ; it is the same as before. Restoring now 
the symbols represented by A, we have for the whole ex- 
pression — 

+ a+ b — c —f— b + a+f— c, 

which, by cancelling {Art. 403) and by adding like symbols 
with like signs, reduces to — 

2 a — 2 c. 

To test this result, let the quantity which was represented; 
by A have the proper numerals substituted, thus : 

+ a-^b — c —/, 



402 ALGEBRA. 

The sum of the given quantity required to be subtracted 
was before found to amount to i^^, therefore — 

A-ii 
becomes — 

3^-11=: 2. 

And the result by the symbols as above was — 

2 a — 2 c, 
which becomes — 

2X2 — 2X1, 

or — 

4 — 2 = 2; 

a result the same as before, proving the work correct. An 
examination of the signs in the above expression, which de- 
notes the problem performed, will show that the sign of each 
symbol which was required to be subtracted has been 
changea in the operation of subtraction. Before subtract- 
ing they were — 

(+/^ — a—/+c); 

after subtraction they are — 

{— d + a-rf— c). 

By this result we learn, that to sttbtract a quantity we have 
but to change its sign and annex it to the quantity from 
which it was required to be subtracted. 

Example : Subtract a — b from c + d. Answer, c + d— a\ b. 

If numerals be substituted, say a— J,b— a,, c— ^, and 
d—<^, then — 

c + d becomes 5+9= 14, 
a-b - ;_4=3, 



So, also, — 
becomes — 



c ^ d — {a — b') ^= 14 ■ 
c + d — a-\- b 

5+9-7 + 4 = 



FRACTIONS ADDED AND SUBTRACTED. 403 

405. — Algebraic Fractions: Added and Subtracted. — 

When algebraic fractions of like denominators are to be 
added or subtracted, the same rules {Arts. 385 and 386) are 
to be observed as in the addition or subtraction of numeri- 
cal fractions — namely, add or subtract the numerators for a 
new numerator, and place beneath the sum or difference the 
common denominator. 

For example, what is the sum of -7,-7,-7? 

000 

For this we have — 

~b * 
Subtract -j from ^. For this we have — 

b-c 
d * 

What is the algebraical sum of — 
be 1^ J 

1: d^ - d^ ^"^ 

For these we have — 

b + c — 7t — r 
~d • 

To exemplify this, let b represent 9, ^ = 8, ;/ = 2, r = 3, 
and d= 12. 

Then, for the algebraic sum, we have — 

9 + 8 — 2 — 3 _ 12^ _ 

12 ~ \2~ ^' 

Now, taking the positive and negative fractions sep- 
arately, we have — 

-9 , A. ^ iZ . 
12 12 12 ' 
and — 

-2 -3^-5 
12 12 12 ' 



r? 



404 ALGEBRA 

Tog-ether — 



i; -5 ^ 12 ^ ^ 

12 12 12 ' 



as before. 



406, — The Lieast Common Denominator. — When the 
denominators of algebraic fractions differ it is necessary be- 
fore addition or subtraction can be performed to harmonize 
them, as in the reduction of the denominators of numerical 
fractions {Arts. 388-390). For example, add together the 

tractions v— , y, — . In these denommators we perceive 

that they collectively contain the letters a, b and c, and no 
others. It will be requisite, therefore, that each of the frac- 
tions be modified so that its denominator shall have these 
three factors. To effect this it will be seen that it is neces- 
sary to multiply each fraction by that one of these letters 
which is lacking in its denominator. Thus, in the first, a is 

11- 1 r r ^ ^ a yi a =^ a a ^ . . 

lacking, therefore {Art. 380) j— _ -y-. In the second a 

(J C y^ CL ' CI' C/ 
€ y^ Ct C — CL C € 

and c are lacking, therefore j _ —7-, and in the third 

T X b ^=^ T b 
b is lackinsr, therefore — r —r- Placing them now 
^' ac y. b — abc ^ 

together we have — 

aa + ace + br _ a e r 
a be. ~ b c b ac 

The factor a a may be represented thus a", which means 
that a occurs twice, the small figure at the top indicating 
the number of times the letter occurs ; ^' is called a squared, 
a a a = a^y and is called a cubed. 

In order to show that the above fraction, resulting as the 
sum of the three given fractions, is correct, let ^ = 2, ^ = 3, 
c = 4, ^ = 5, and r = 6. Then the three given fractions 
are — 



3x4 3 2x4 6 3 4 



FRACTIONS SUBTRACTED. 405 

In equalizing these denominators we multiply the second 
fraction by 2, and the third by i^^, which will give — 



then- 



Now the sum of the fractions is — 

a"^ + ace + br 
ab c 







5 


X 2 = 


ID 


3 X 


li 




4* 










3 


X 2 = 


6 


'4x 


I* 




6 


' 




I 


. 


ID 


4* 


I 


:^*' 




^* 






7 


6 


+ 


6 


-^6 ■ 


— ~ 


6 ~ 


: 2 


6 


= 


2 


12' 



or. 



2^ + 2x4x5 + 3x6 
2x3x4 



4 + 40+18 62 14 7 

or, =:^ ■ z=: 2 = 2 ! 

24 24 24 12 ' 

the same result as before, thus showing that the reduction 
was rightly made. 

407- — Algebraic Fractions Subtracted. — To exemplify 
the subtraction of fractions, let it be required to find the 

algebraic sum of - — -> — -p. These denominators all dif- 
cdf 

fer. The fractions, therefore, require to be modified, so 
that each denominator shall contain them all. To accom- 
plish this, the first fraction will need to be thus treated : 





a X df= adf 


the second — 


c X df= cdf 




bxcf= bcf 


the third— 


dxcf= cdf 


The sum of these is- 


e X c d = cde 
f X c d - cdf 




adf — bcf— cde 



cdf 



406 ALGEBRA. 

That this is a correct answer, let the result be proved by 
figures ; thus, for ^ put 15 ; ^, 2 ; <:, 3 ; ^,4; ^, 5 ; /, 6. Then 
we shall have — 

a b ^ _ 15 2 5 
c ^~'7~3~4~6* 

It will be observed that these denominators may be equal- 
ized by multiplying the first fraction by 2, and . the second 
by \\, therefore we have — 

JO _ 3 _ 5. 
6 ^ ^' 

To make the required subtraction we are to deduct from 30 
(the numerator of the positive fraction), first 3, then 5 ; or, 
the sum of the numerators of the negative fractions ; or for 
the numerator of the new fraction we have 30 — 8 = 22. 
The required result, therefore, is — 

6-3-38. 
To apply this test to the algebraic sum we have — 



a d f — bcf— cde_ 15x4x6 + 2x3x6+3x4x5 
c df ~ 3x4x6 ' 

which by multiplication reduces to — 

360 — 36 — 60 264 22 1 1 



72 72 6 3 



= 31 



a result the same as before, proving the work correct. An- 
other example : 

From take -, — and - ; 

71 VI n vt n 

or, find the algebraic sum of— 

a ,b c d e 

n ni 71 m ti 



a 


c 




e a — c — e 


n 


n 




n ~ n 




b 




d b-d 






— 


z=. 




7)1 




m m 



DENOMINATORS HARMONIZED. 407 

The fractions which have the same denominator may be 
grouped together thus : 



and- 



To harmonize these two denominators, m and n^ the first 
fraction must be multiplied by vi and the last by n, or — 

m(a — c — e) n {b — <:/) _ in {a — c — e) + n {b —d) 
7n n ' vi n ~ m n 

In the polynomialfactor within the parentheses (^ — c — e)viQ 
have the positive quantity a, from which is to be taken the 
two negatives c and ^, or their sum is to be taken from a^ or 
{a — {c ^ e) ). With this modification we have for the alge- 
braic sum of the five given fractions — 

m {a — {c + c)) + n {b — d) 
mn 

To test the accuracy of this result, let the value of the sev- 
eral letters respectively be as follows: ^ = ii,/^ = 9, c = 3^ 
^ = 4, ^ = 5, ?;^ = 10, and n = 8. Then the sum is — 

10(11 -(3 + 5)) + 8(9-4) ^70^7 
10 X 8 80 8* 



n n n 8 \8 8/ 8 8 ~ 8 



Now, taking the fractions separately, we have — 

\8 ^ 87 8 

b d _ g 4 5 

' m m ~ 10 ~ ~io ~ To' 

or, together we have, as the sum of these two results- 

8 ^ 10 



4o8 



ALGEBRA. 



To harmonize these denominators we may multiply the first 
fraction by 5, and the second by 4, thus: 

8 X 5 = 40' 10 X 4 =40' 
and then the sum is — 

il ^ _ il — Z. 
40 "^ 40 ~" 40 ~ 8 ' 

the same result as before, thus the accuracy of the work is 
established. 

4-08, — Graphical Representation of Multiplication. — 

In Fig, 278, let ABCD,2i rectangle, have its sides A B and 




























































Fig. 278. 

A C divided into equal parts. Then the area of the figure 
will be obtained by multiplying one side by the other, or 
putting a for the side A By and b for the side A C, then the 
area will be a x b, or a b. This will be the correct area of 
the figure, whatever the length of the sides may be. If, as 
shown, the area be divided into 4 x 7 == 28 ^ual rectangles, 
then a would equal 7, and b equal 4, and ^ ^ = 7 x 4 = 28, the 
area. \i A B equal 28 and A C equal 16, then will a = 28, 
and b = 16, and a b = 2S x 16 = 448, the area. 

4-09. — Graphical Multiplication : Three Factors. — Let 

A B CD E FG {Fig. 279) represent a rectangular solid which 
may be supposed divided into numerous small cubes as 
shown. Now, if a be put for the edge A B^b for the edge 
A C, and c for the edge CD, then the cubical solidity of the 



MULTIPLICATION OF A BINOMIAL. 



409 



whole figure will be represented hy axbxc = abc. If the 
edge A B measures 6, the edge A C '^, and the edge (7i> 4, 
then ^^^==6x3x4— 72 = the cubic contents of the 
figure, or the number of small cubes contained in it. 





A 










B 


////// /\ 


c 


/ 


/ / / / / 


/ 

/ 
/ 

/ 

/ 


////// y 










' 


V 














/ 














/ 


li^ 












/ 


V 


D 












£ 





Fig. 279. 

4-10. — Graphic Repre§ei]tatioii : Two and Three Fac- 
tors. — Figs, 278 and 279 serve to illustrate the algebraic ex- 
pressions a b and a b c. In the former it is shown that the 
multiplication of two lines produces a rectangular surface, 
or that if a and b represent lines, then a b may represent a 
rectangular surface {Fig. 278) having sides respectively 
equal to a and b. And so if a, b^ and c represent three sev- 
eral lines, then ab c may represent a rectangular solid {Fig. 
279) having edges respectively equal to a, b^ and c. 


































































C 



Fig. 280. 



411. — Graphical Multiplication of a Binomial. — Let 

A B C D {Fig. 280) be a rectangular surface, and B E D F an- 
other rectangular surface, adjoining the first. The area of 
the whole figure is evidently equal to — 

{AB + BE)-xAC. 



^^O Ai^GEBKA. 

The area is also equal to — 



ABxAC + BExBD', 
or, since A C = B D, the area equals — 



ABxAC + BExAC\ 

or, if symbols be put to represent the lines, say a for A B, 
b for B E, and c for A Cy then the two representatives of the 
area, as above shown, become : The first — 

{a Jr b) xc =z area ; 
and the last — 

{ax c) j^ {b xc)^= area. 
Hence we have — 

{a + b) c ^ a c + b c. 

This result exemplifies the algebraic multiplication of a bi- 
nomial, which is performed thus : Let ^ + ^^ be multiplied 
by c. 
The problem is stated thus : 

{a + b) c. 

To perform the multiplication indicated we Droceed thus : 

a -^ b 
c 



ac -\- b c 



multiplying each of the factors of the multiplicand sepa- 
rately and annexing them by the sign for addition. Putting 
the two together, or showing the problem and its answer in 
an equation, we have — 

(aA-b^c^ac-vbc, 

producing the same result, above shown, as derived from 
the graphic representation. 

4(2. — Graphical ISquaring^ of a Binomial. — Let EG C J 

{Fig. 281) be a rectangle of equal sides, and within it draw 



SQUARING OF A BINOMIAL 



411 



the two lines, A H and F D, parallel with the Unes of the 
rectangle, and at such a distance from them that the sides, 
A B and B D, of the rectangle, A B CD, shall be of equal 
length. We then have in this figure the three squares, 
E G C y, A B CD, and FGBH, also the two equal rect- 
angles, EFAB and BHD J. 

Let F Fhe represented by a and FGhy b, then the area 
of A B C D will be axa — a"" \ the area of FG B H will be 
b xb = b^' \ the area of EFAB will hQ ax b = ab, and that 



• 




A B 


\ 


C 1 


[) J 



Fig. 281. 

of BHD y will be the same. Putting these areas together 
thus — 

a"" ■\-2ab ^b"", 

the sum equals the area of the whole figure — equals the prod- 
uct of EG X E C — equals the product — 



{a-^b) x{a + b). 
So, therefore, we have — 

{a + b) {a + b) = a' + 2 a b + b"" ] 



(112.) 



or, in general, the square of a binomial equals the square of 
the first, plus twice the first by the second, plus the square of the 
second. This result is obtained graphically. The same re- 
sult may be obtained by algebraic n;ultiplication, combining 



412 ALGEBRA. 

each factor of the multiplier with each factor of the multi- 
plicand and adding the products, thus — 

a -^ b 
a ■\- b 



ab + b' 



a"" ^ 2 ab + b\ 



The same result as above shown by graphical representa- 
tion. 

413. — Graphical Squaring of tlie Difference of Tivo 
Factors. — Let the line E C {Fig. 281) be represented by c, 
and the line A E and A C 2ls before respectively by 3 and 
a, then — 

EC-AE = AC. 

c — b ^^ a. 

From this, squaring both sides, we have — 

[c-by = a\ 

The area of the square A B C D may be obtained thus : 
From the square E G CJ take the rectangle E G x E A and 
the rectangle F G x D J, minus the square F G B H, or 
from c"^ take the rectangle cb, and the rectangle c b^ minus 
the square, b ^ and the remainder will be the square, a ^ ; or, 
in proper form — 

c^ — c b — c b ^ b"" = a"^ 

In deducting from c"" the rectangle cb twice, we have taken 
away the small square twice ; therefore, to correct this 
error, we have to add the small square, or b I Then, when 
reduced, the expression becomes — 

c' -2cb-^b'' = a''^{c-b)\ 
This result is obtained graphically. The result by algebraic 



PRODUCT OF THE SUM AND DIFFERENCE. 



413 



process will now be sought. The square of a quantity may 
be obtained by multiplying the quantity by itself, or— 



c - b 
c- b 

> - be 
-bc-vb' 



{c-by=:c'-2bc + b\ 



(113.) 



In this process, as before, each factor of the multiplier is 
combined with each factor of the multipHcand and the sev- 
eral products annexed with their proper signs {Art. 415), 
and thus, by algebraic process, a result is obtained precisely 
like that obtained graphically. This result is the square of 
the difference of c and b ; and since c and b may represent 
any quantities whatever, we have this general — 

Rule. — The square of the difference of two quantities is 
equal to the sum of the squares of the tzvo quantities^ minus twice 
their product. 







E H 


1^ 



Fig. 282. 



4.(4, — Graphical ProdM<«t ol* the Sum and Differenc« 
of Two Quantities. — Let the rectangle A B C D {Fig. 282) 
have its sides each equal to a. Let the line E F hQ parallel 
with A B and at the distance b from it, also, the line F G 
made parallel with B D, and at the distance b from it. Then 
the line E F equals a -\- b, and the line E C equals a — b. 
Therefore the area of the rectangle E F C G equals a -f /;, 



414 ALGEBRA. 

multiplied hy a — b. From the figure, for the area of this 
rectangle, we have — 

EFCG\ 



ABCD 


-ABEH+HFDG 


or, by substitution of the symbols, 






a"" — a b + b {a - 


-b). 


Multiply the last 


quantity thus — 

a-b 
b 





ab-b' =b{a^b). 

Substituting this in the above we have — 

a'' — ab ^ ab — b"" = { a-r b) x {a — b). 

Two of these like quantities, having contrary signs, cancel 
each other and disappear, reducing the expression to this — 

a'-b' = {a-{-b)x{a-b). 

The correctness of this result is made manifest by an inspec- 
tion of the figure, in which it is seen that the rectangle E FC G 
is equal to the square ABCD minus the square BJHF. 
For ABEH equals BJDG. Now, if from the square 
A B C D wQ take away A B E H^ and place it so as to cover 
BJDG, we shall have the rectangle EFCG plus the square 
BJHF] showing that the square ABCD is equal to the 
rectangle EFCG plus the square BJHF] or — 

a"" ={a + b) X {a — b) + b\ 

The last quantity may be transferred to the first member of 
the equation by changing its sign {Art. 403). Therefore — 

a-" -b'' = {a + b)x {a-b), 
as was before shown. 



MULTIPLICATION— PLUS AND MINUS. 



41 



The result here obtained is derived from the geometrical 
figure, or graphically. Precisely the same result may be 
obtained algebraically ; thus — 

a -v b 
a — b 



a -\-ao 
— ab 



{a + b) X {a — b) = a' 



(114.) 



Here the two like quantities, having unlike signs, cancel 
each other and disappear, leaving as the result only the dif- 
ference of the squares. 

The result here obtained is general ; hence we have this — 

Rule. — TJie product of the sum and differe^ue of two quan- 
tities equals the difference of their squares. 



Fig. 283. 

415- — Plus and minus Sig^ns in multiplication. — In pre- 
vious articles the signs in multiplication have been given to 
products in accordance with this rule, namely: Like signs 
give plus ; unlike signs^ minus. This rule may be illustrated 
graphically, thus : In the rectangular Fig. 283, let it be re- 
quired to show the area of the rectangle A G C H, in terms 
of the several parts of the whole figure. Thus the area of 
AGE7 qq^utA ABEF-GB7F2Lnd the area of EJCH 
equals EFCD - J F H D. And the areas of AGEJ^ 
E JCH equals the area of ^ 6^ CH. Therefore the sum of 
the two former expressions equals AG CH. Thus ABEF— 
GBJF+EFCD-JFHD^AGCH. Let the several 
lines now be represented bv algebraic symbols; for example, 



4i6 



ALGEBRA. 



\^t AB=^EF=a', \QtGB = yF=b', \tt A E =^ G J = c ■ 
and E C ^^ J H = d, and let these symbols be substituted for 
the lines they represent, thus A BEE— G BJ F ^ E EC D — 
JFHD^AGCH. 



ac 



b c + ad — b d ^^ ij^ — ^) x {c -\- d). 



An inspection of the figure shows this to be a correct 
result. It will now be shown that an algebraical multiplica- 
tion of the two binomials, allotting the signs in accordance 
with the rule given, will produce a like result. For example — 



a — b 
c + d 



a 



c — b c -\- ad 



d. 




Fig. 284. 



4-16. — Equality of Squares on Oypotlienusc and j^icle§ of 
Right-Angled Triangle. — The truth of this proposition has 
been proved geometrically in Art. 353. It will now be 
shown graphically and proved algebraically. 

Let A BCD {Fig. 284) be a rectangle of equal sides, and 
BED the right-angled triangle, the squares upon the sides 
of which, it is proposed to consider. Extend the side BE 
to F\ parallel with BE draw Z>6^, CK, and A L. Parallel 
with ED draw A J and L G. These lines produce triangles, 
AHB, AC J, ALC, CKD, and CGD, each equal to the 
given triangle BED {Art. 337). Now, if from the square 



SQUARES ON RIGHT-ANGLED TRIANGLE. 417 

A B C D V7e take A ^// and place \t:itCDG\ and if we take 
BED and place it at ALC we will modify the square 
A BCD, so as to produce the figure LG D E HA L, which 
is made up of two squares, namely, the square D E EG and 
the square ALEH, and these two squares are evidently 
equal to the square A B CD. Now, the square DE EG is the 
square upon ED, the base of the given right-angled triangle, 
and the square A L EH is the square upon A H ^^ B E, the 
perpendicular of the given right-angled triangle, while the 
square A B C D is the square upon B D, the hypothenuse of 
the given right-angled triangle. Thus, graphically, it is shown 
that tJie square iipon the hypothenuse of a right-angled triangle 
is equal to the stun of the squares up07i the remaining two sides. 
To show this algebraically, let B E, the perpendicular of 
the given right-angled triangle, be represented hy a\ E D, 
the base, b}^ b, and B D, the hypothenuse, by c. Then it is 
required to show that — 

e' = a'+b\ 

Now, since D K ^ B E ^^^ a, therefore, E K ^= E D — 
D K = b — a, and the square ^ A" y 77 equals (b — «)^ which 
{Art. 413) equals 

b"" — 2ab + a^. 

This is the value of the square E KJ H which, with the four 
triangles surrounding it, make up the area of the square 
A B C D. Placing the triangle A B H oi this square outside 
ot it at CD G, and the triangle B E D "dX A L C, we have the 
four triangles, grouped two and two, and thus forming the 
two rectangles C G D K and A L CJ. Each of these rect- 
angles has its shorter side {A L, C G) equal to B E — a, and 
its longer side L C, G D, equal to E D — b ; and the sum of 
the two rectangles is ab ^ ab =. 2ab. This represents the 
area of the two rectangles, which are equal to the four tri- 
angles, which, together with the square EKJH, equal the 
square A BC D\ or — 

ABCD^EKyn^-CGDK+ALCJ, 



4l8 ALGEBRA. 

or — c"" — {b — of ^ a b ^ a b, or— 

c''^\b-ay-\-2ab. 

Then, substituting for {b — of, its equivalent as above, we 
have — 

c""— b""— 2ab^a''-\-2ab, 

Remove the two like quantities with unlike signs {Art. 402), 
and we have — 

c'^ =^ b' -r a' \ (115.) 

which was to be proved. 

417. — I>ivi§i©ii the Reverse of MuUiplieatiota. — As di- 
vision is the reverse of multiplication, so to divide one quan- 
tity by another is but to retrace the steps taken in multipli- 
cation. If we have the area ab {Fig. 278), and one of the 
factors a given to find the other, we have but to remove 
from a b the factor a, and write the answer b. 

If we have the cubic contents of a solid abc {Fig. 279), 
and one of the factors a given to find the area represented 
by the other two, we have but to remove a, and write the 
others, be, as the answer. 

If there be given the area represented by a {b^c) (see 
Art. 411), and one of the factors a to find the other, we have 
but to remove a and write the answer b^ c. Sometimes, how- 
ever, a {b + c) is written ab + ac. Then the given factor is 
to be removed from each monomial and the answer written 
b + c. 

If there be given the area represented hy a"^ + 2 a b + b"" 
to find the factors, then we know by Art. 412 that this area 
is that of a square the sides of which measure a + b, and that 
the area is the product of <^ + ^ hy a + b \ or, that a + b is 
the square root oi a^ + 2 ab + b"'. 

• If there be given the area a^ — 2 a b + b~ to find its fac- 
tors, then we know by Art. 413 that this area is that of a 
square whose sides measure a — b, or that it is the product 
o^ a — b by. a — b, or :the square of ^ — b. 



PROCESSES IN DIVISION. 419 

If there be given the difference of the squares of two 
quantities, or the area represented hy a- — b"-, to find its fac- 
tors, then we know by Art. 414 that this is the area pro- 
duced by the multiplication oi a — bhy a + b. 

418. — I>ivi§ion : Statement of Quotient. — In any case of 
division the requirement may be represented as a fraction ; 
thus : To divide c + d — fhy a — b we write the quotient 
thus — 

c + d — f 



For example, to illustrate by numerals, let <^ = 7, (5 = 3, 
<: = 4, <r/ = 5, and/ = 6. Then the above becomes — 

4+5-6 ^ ^ 
7-34 

4 1 9. — Division ; Reduction. — When each monomial in 
either the numerator or denominator contains a common 
quantity, that quantity may be removed and placed outside 
of parentheses containing the monomials from which it was 
taken ; thus, in — 

2 ab + d^ ac — % ad 

7 ' 

we have 2 and a factors common to each monomial of the 
numerator. Therefore the expression may be reduced to 

2a{b^2c — \d) 

7 • 

To test this arithmetically we wilU put <3; = 9, ^ = 7, ^=5, 
^ = 4, and f =z 6, Then for the first expression we have — 



2x9x7 + 4x9x5 — 8x9x4 
6 ' 

which equals — 

126 + 180 — 288 
7. = 3. 



420 ALGEBRA. 

And for the second expression — 



2x9 (7 + 2x 5— 4x4) 
6 
which equals — 

18 (17 -^ 16) 18 

6 - 6 "~ ^' 

the same result as before. It will be observed that in this 
process of removing all common factors algebra furnishes 
the means of performing the work arithmetically with many 
less figures. The reduction is greater when the common 
factors are found in both numerator and denominator. For 
example, in the expression — 

'^ an -\- (^ bii — \^ en 
12 dn — i^ fn 

we have 3 n a factor common to each monomial in the nu- 
merator and denominator ; therefore the expression reduces 

to 

Zn{a + ^ b - S c) 
Zn{A,d-6f) 

And now, since 3 ;2 is a factor common to both numerator 
and denominator, these cancel each other ; therefore (Art. 
371) the expression reduces to — 

a + lb — ^c 



4.d - 6f ' 



To test these reductions arithmetically, let ^ = 9, <^ = 8, 
r = 4, ^ = 6, /=: 3, and n = ^. Then the first expression 
becomes — 



which equals — 



3x9x5+9x8x5- 15 x4x5 
12x6x5— 18x3x5 

135 + 360 - 300 ^ 295 _ ^ 1 . 
360 — 270 ~ 90 6 ' 



FORMULA OF THE LEVER. 42 1 

and the second expression becomes — 



9 + 3x8 — 5x4 
4x6 — 6x3 * 

which equals — 



+ 24 - 20 ^ 23 ^2^ 



24—18 6 6 

The same result, but with many less figures. 

420. — Proportionals : Analysis. — In the formula of the 
lever {Art. Z77)j P x CF —Ry.EC. Let n be put for the 
arm of leverage (T/^and m for E C. Then we have — 

Pn = Rfftj 
from which by division {Art. 372) we have {Art. 399) — 

P-i?— , (no.) 

':'"^- R = p^, (III.) 

Suppose there be a case in which neither R nor P severally 
are known, but that their sum is known ; and it is required 
from this and the m and n to find R and P. Let — 

W= R + P, 

then— W - R = P. (See Art. 403.) 

The value of P was above found to be — 



p=r'-^ 



111 

Since P — R — and also equals W — R, therefore- 



m 
W-R= R— 
n 



422 ALGEBRA. 

Transferring R to the opposite member {Art. 403) we have — 



W:=: R + R—. 
n 



Here R appears as a common factor and may be separated 
by division {Art. 419); thus — 



W 



= ^(-^)- 



By division the factor ( i + — j may be transferred to the 
opposite member {Art. 371). Thus we have — 

• W 

n 

by which we find the value of R developed. As an example, 
let W = 1000 pounds, 7n = 3 feet and ?2 = 7 feet ; then — 

_ 1000 _ 1000 

1 -t- 7 7 

Multiplying the numerator and denominator by 7, we get — 

7 X 1000 

R — = 700. 

10 ' 

Since — . R + P = 1000, 

and — R = 700, 

then — P = 300. 

But a process similar to the above develops an expression 
for the value of P, which is — 



P= »- 



I + Jt 
m 



("7.) 



Putting this to the test of figures, we have — 



000 1000 3000 



_, l\J\^<<J iwv^w -sv^w^-/ 

^=^^ = ^r = ^7r = 3oo. 



NEGATIVE EXPONENTS. 423 

4-21. — Rai«^iii^ a Quantity to any IPower. — When a 
quantity is required to be multiplied by its equal, the prod- 
uct is called the square of the quantity. Thus a y. a ^=^ d' 
{Art. 412). If the square be multiplied by the original 
quantity the result is a cube ; or, ^^'^ x ^ = <^' ; or, generally, 
for — 

a^ a a, a a a, aaaa, aaaaa^ 
we put — 

a, <?^ rt:', a\ d" \ 

in which the small number at the upper right-hand cor- 
ner indicates the number of times the quantity occurs in 
the expression. Thus, if a ~ 2, then rt;" = 2 x 2 = 4, 
^?' = 4.x 2 = 8, rt:* = 8 X 2 = 16, rt' = 16 X 2 = 32 ; any term 
in the series of powers may be found by multiplying the 
preceding one by a, or by dividing the succeeding one by a. 

Thus a" y. a ■=. a', and ~^^ a^. 
a 



422. — Quantities with XegaSive Exi)onent§. — The series 
of powers, by division, may be extended backward. Thus, 

11 we divide -^ a \ —=^ a ; -— a \ — ^= a \ - = a \ -^ = a ; 
a a a a a a 

— z^ a \ — = a \ etc. 
a a 

In this series we have - =■ a\ But a quantity divided by 

its equal gives unity for quotient, or ^ = i . Therefore, - = i , 

3 ^ 

and rt:° = I. This result is remarkable, and holds good re- 
gardless of the value of a. 

From this and the preceding negative exponents we de- 
rive the following : 



"iP^>^T^^S^ 



424 ALGEBRA. 

a =- = I, 

a 

a'=-= i. 
a a' 

a a a a^' 

3 a~' I I ^ 

<^ =- =-^r- = -:, etc. 

a a a a 

Showing that a quantity with a negative exponent may have 
substituted for it the same quantity with a positive exponent, but 
used as a denominator to a fraction having unity for tJie 
numerator. 

4-23. — Addition and Subtraction of Exponential Quan- 
tities. — Equal quantities raised to the same power may be 
added or subtracted; as, cr -\- 2 <^^ = 3^;^; but expressions in 
which the powers differ cannot be reduced ; thus, a^ ^^a — a" 
cannot be condensed. 

424. — HuBtipHcation of Exponcntia3 Quantities. — It 

will be observed in Art. 421 that in the series of powers, the 
index or exponent increases by unity ; thus, d, d^, a^, a", etc. ; 
and that this increase is effected by multiplying by the root, 
or original quantity. From this we learn that to multiply 
two quantities having equal roots we simply add their exponents. 
Thus the product of a^ d\ and d^ is a' x d^ y^ a"" — a\ 
The product of a~', <^^ and a"" is a"^ x a^ x d" =^ a\ 
The exponents here, are : — 2 + 3 + 5=8 — 2 = 6. 

425. — I^ivision of Exponential Quantities. — As division 
IS the reverse of multiplication, to divide equal quantities 
raised to various poivers, we need simply to subtract the expo- 
nent of the divisor from that of the dividend. Thus, to divide 
a^ by a^ we have d"^ = a^. That this is correct is manifest ; 
for the two factors, d^ x a^, in their product, ^^ produce the 
dividend. 

To divide ci by a", we have a^''" = ^"^ which is equal to - 

a" 



EXPLANATION OF LOGARITHMS. 4^5 

(see Art. 422). The same result may be had by stating the 
question in the usual form. Thus, to divide a" by a" we have 

- , a fraction which is not in the lowest terms, for it may be 
a 

2 2 

put thus, -^-^ = — , by which it is seen that it has in both its 
a a a 

numerator and denominator the quantity a\ which cancel 
each other {Art. 371). Therefore, -, = - ; the same result 
as before. 

4-26. — Extraction of Radicals. — We have seen that the 

square of a \s a^ x a^ ^= a^ \ oi 2 a'^ is 2 a"^ x 2 a^ = 4 a'' ; in each 

case the square is obtained by doubling the exponent. 

To obtain the square root the converse follows, namely, 

take half of the exponent. 

Thus the square root of a'^ is a\ of a^ is a, of a^ is a\ 
The same rule, when the exponent is an odd number, 

gives a fractional exponent, thus : the square root of a^ is a^ ; 

or, of a\ is ai. So, also, the square root of a, or a\ is ai- 

Therefore, we have a^ = Va, equals the square root of a, and 

the cube root of a^ z= ai ^= Va. 

427. — E.ogaritlQm§. — We have seen in the last article the 
nature of fractional exponents. Thus the square root of a^ 
equals a^, which may be put a'^^. In this way we may have 
an exponent of any fraction whatever, as a^k Between the 
exponents 2 and 3, we may have any number of fractional 
exponents all less than 3 and more than 2. So, also, the 
same between 3 and 4, or any other two consecutive num- 
bers. 

The consideration of fractional exponents or indices has 
led to the making of a series of decimal numbers called 
logarithms, Avhich are treated in the manner in which expo- 
nents are treated ; namely — 

To multiply numbers add their logarithms. 

To divide numbers, subtract the logarithm of the divisor from 
the logarithm of the dividend. 



426 ALGEBRA. 

To raise any number to a given pozver^ multiply its logaritJim 
by the exponent of that poiver. 

To obtain the root of any poiver^ divide the logarithm of the 
given number by the exponent of the given power. 

As an example by which to exemplify the use of loga- 
rithms : What is the product of 25 by 375 ? 

We first make this statement : 

Log. of 25- = I- 

- 375. =2. 

Putting at the left of the decimal point the integer char- 
aeteristic, or whole number of the logarithm at one less than 
the number of figures in the given number at the left of its 
decimal point. 

To find the decimal part of the required logarithm we 
seek in a book of Logarithms (such as that of Law's, in 
Weale's Series, London) in the column of numbers for the 
given number 25, .or 250 (which is the same as to the man- 
tissa) and opposite to this and in the next column we find 
7940 and a place for two other figures, which a few lines 
above are seen to be 39 ; annex these and the whole number 
is 0-397940. These we place as below : 

Log. of 25. = 1.397940. 

Now, to find the logarithm of 375, the other factor, we 
turn to 375 in the column ol numbers and find the figures 
opposite to it, 4031, which are to be preceded by 57-, the two 
figures found a few lines above, making the whole, -574031, 
which are placed as below, and added together. 

Log. of 25. = 1-397940 

'' 375- = 2-574 031 

The sum — 3-971971 

This sum is the logarithm of the product. To find the 
product, we seek in the column of logarithms, headed o-,, 
for -97 1 97 1, the decimal part. We find first 97, the first two 



EXAMPLES OF LOGARITHMS. 427 

figures, and a little below seeking for 197 1, the remaining 
four figures, we find 1740, those which are the next less, 
and opposite these, to the left, we find 7, and above 93, or 
together, 937 ; these are the first three figures of the required 
product. 

For the fourth figure we seek in the horizontal column 
opposite 7 and 1740 for 1971, the remaining four figures of 
the logarithm, and find them in the column headed 5. 

This figure 5 is the fourth of the product and completes 
it, as there are only four figures required when the integer 
number of the logarithm is 3. The completed statement 
therefore is — 

Log. of 25. = 1.397940, 
" " 375- = 2.574031, 



" " 9375 = 3-971971. 

Another example in the use of logarithms. What is the 
product of 3957 by 94360? 

The preliminary statement, as explained in last article, is — 

Log. 3957 = 3. 
'' 94360 = 4. 

In the book of logarithms seek in the column of numbers 
for 3957. In the first column we find only 395, and opposite 
to this, in the next column, we find a blank for two figures, 
above which are found 59. Take these two figures as the 
first two of the mantissa, or decimal part of the required 
logarithm, thus, 0-59. Again, opposite 395 and in the col- 
umn headed by 7 (the fourth figure of the given number), 
we have the four figures 7366. These are to be annexed to 
(o. 59) the first two obtained. The decimal part of the loga- 
rithm, therefore, is 0-597366. 

To obtain the logarithm for 94360, the other given num- 
ber, we proceed in a similar manner, and, opposite 943, we 
find 0-97; then, opposite 943 and in column headed 6, we 
find 4788, or, together, the logarithm is 0-974788. The 
whole is now stated thus — 



428 ALGEBRA. 

Log. of 3957 = 3-597366 

" " 94360 = 4.974788 



" '' 373382000 = 8-572154 = sum of logs. 

The two logarithms are here added together, and their sum 
is the logarithm of the product of the two given factors. 
The number corresponding to the above resultant logarithm 
may be found thus: Look in the column headed o for 57, 
the first two numbers of the mantissa, then in the same 
column, farther down, seek 2154, the other four figures of 
the mantissa; or, the four (1709) which are the next less 
than the four sought, and opposite these to the left, in the 
column of numbers, will be found 373, the first three figures 
of the product ; opposite these, to the right, seek the four 
figures next less than 2154, the other four figures of the man- 
tissa. These are found in the column headed 3 and are 
2058. The 3 at the head of the column is the fourth figure 
in the product. From 2154, the last four figures of the man- 
tissa, deduct the above 2058, or — 

2154, 
2058, 



Remainder, 96. 

At the bottom of the page, opposite the next less number 
(3727) to that contained in 3733, the answer already found, 
seek the number next less to the above remainder, 96. This 
is 92-8, and is in the column headed 8. Then 8 is the next 
number in the product. From 96 deduct 92-8, and multi- 
ply it by 10, or — 

96 

92-8 

3-2 X 10 =ri 32. 

Then, in the same horizontal column, seek for 32 or its next 
less number. This is 23-2, found in column 2. This 2 is the 
next figure in the product. Additional figures may be ob- 
tained by the table of proportional parts, but they cannot be 



THE SQUARE OF A BINOMIAL. 429 

depended upon for accuracy beyond two or three figures. 
We therefore arrest the process here. 

The product requires one more figure than the integer of 
the logarithm indicates; as the integer is 8, there must be 
nine figures in the product. We have already six ; to make 
the requisite number nine we annex three ciphers, giving 
the completed product — 

3957 X 94360 = 373382000. 

By actual multiplication we find that the true product in the 
last article is 373382520. In a book of logarithms, carried to 
seven places, the required result is found to be 373382500^ 
which is more nearly exact. 

The utility of logarithms is more apparent when there 
are more than two factors to be multiplied, as, in that case, 
the operation is performed all in one statement. Thus : 
What is the product of 3-75, 432-95, 1712, and 0-0327 ? 

The statement is as follows : 

Log. 3-75 == 0-574031 

432-95 = 2-636438 

1712- = 3-233504 

•032 7 = 8-51454 8 

Product = 90891. = 4-958521 

16 

5. 

Explanations of working are given more in detail in most of 
the books of logarithms. 

4-28. — Completing the Square of a Binomial. — We 

have seen in Art. 412 that the square of a binomial {a + b) 
equals a"" + 2 ab ^ b"" — a trinomial — the first and last terms 
of which are each the square of one of the two quantities, 
Avhile the second term contains the second quantity multi- 
plied by twice the first quantity — 

In analytical investigations it frequently occurs that an 
expression will be obtained which may be reduced to this 
form : 



430 ALGEBRA. 

a"^ + m a b =^ f, (i i8.) 

in which m is the coefficient of the second term, and a and b 
are two quantities represented by a and b or any other two 
symbols. 

A comparison of this expression with the square of a bi- 
nomial (ii2.) contained in Art. 412, shows that the member 
at the left comprises two out of the three terms of the square 
of a binomial ; as thus — 

a^ + 2 a b Jr ^"^y 

but with a coefficient m instead of 2. It is desirable, as will 
be seen, to ascertain a proper third term for the given ex- 
pression ; or, as it is termed, '' to complete the square." The 
method by which this is done will now be shown. 

A consideration of the above trinomial shows that the 
third term is equal to the square of the quotient obtained by 
dividing the second term by twice the square root of the 
first ; or — 



/2 abV _ 
\ 2 a J ~ 



Now a third term to the above binomial, equation (118.), may 
be obtained by this same rule. For example — 



/7n a bV / m bV 



The rule for the third term then is: Divide the sccoiid term 
by twice the square root of the first, and square the quotient. 

As an example, let it be required to find the third term 
required to complete the square in the expression — 

6 n X + AfX'^ -= fy 

in which n and / are known quantities and x unknown. 
Putting it in this form — 

^ x"^ ^ 6 n X = fy 
and dividing by 4, we have — 



BINOMIALS CONTINUED. 43I 

X +-„x = -, 
which reduces to — 



1-2 



3 / 



2 4 

Now applying the above rule for finding the third term, we 
have — 



(*^)=e»y 



2 X 

which is the required third term. To complete the square 
we add this third term to both members of the above re- 
duced expression, and have — 

The member of this expression at the left is the completed 
square of a binomial, the two quantities constituting which 
are the square roots of the first and third terms respectively ; 
or X and | 7i, and we therefore have — 

2 ^4 ^^ 4 

and now taking the square root of both sides of the expres- 
sion, we have — 



3 

X + —n 

4 



1''^ e-" )' 



and, by transferring the second quantity to the right mem- 
ber, we have — 



^ A \A J A ' 



an expression in which x, the unknown quantity, is made to 
stand alone and equal to known quantities. 

The process of completing the square is useful, as has 
Deen shown, in developing the value of an unknown quan- 



432 ALGEBRA. 

tity where it enters into an expression in two forms, one as 
the square of the other. 

As an example to test the above result, let/ = 256 and 
n — 8. Then we have by the last expression for the value 
of ^ — 



= |/^^+(3-x8y-3-x8 

^ ZL VA / A. 



;4 

= 1/64+ 36 _ 6, 

= \/ 100 — 6, ' 

X := 10 — 6 = 4. 

Now this value of x may be tested in the original expres- 
sion — 

6 nx ■{■ 4 ;ir ^ = y, 

for which we have — 

6x8x4 + 4x42=/, 

192 + 64 = / 

256=/; 

the correct value as above. 



PROGRESSION. 

429. — Ariilimetical Progre§sion. — In a series of num- 
bers, as 1,3, 5, 7, 9, etc., proceeding in regular order, in- 
creasing by a common difference, the series is called an 
arithmetical progression ; the quantity by which one num- 
ber is increased beyond the preceding one is termed the 
difference. If d represent the difference and a the first term, 
then the progression may be stated thus — 

Terms— i, 2, 3, 4, 5, 

a, a + d, a ^ 2d, a + -i^ d, a + 4.d, etc. 

The coefficient of d is equal to the number of terms preced- 
ing the one in which it occupies a place. Thus the fifth 
term is ^ + ^d, in which the coefficient 4 equals the number 
of the preceding terms. 

From this we learn the rule by which at once to desig- 



ARITHMETICAL PROGRESSION. 433 

nate any term withoutfinding all the preceding terms. For 
the one hundredth term we should have « + 99 <^, or, if the 
number of terms be represented by 71^ then the last term 
would be represented by — 

I =z a + {n — i) d. (119-) 

For example, in a progression where a, the first term, equals 
I, d the difference, 2, and n, the number of terms, 90, the last 
term will be — 

I — a + {n — \) d = I + (90 — 1)2 — 179. 

Therefore, to find the last term : 

To the first term add the product of the common difference 
into the number of terms less one. 

By a transposition of the terms in the above expression, 
so as to give it this form — 

a = I — (n — \) dy (120.) 

we have a rule by which to find the first term, which, in 
words, is — 

Midtiply the number of terms less one by tlie common differ- 
ence, and deduct the product from the last term ; the remainder 
will be the first term. 

By a transposition of the terms of the former expression 
to this form — 

I — a ^ {n — i) d, 



and dividing both members by {n - i), we have- 

I — a 
~~ n - I ' 



(121.,). 



which is a rule for the common difference, and which, ini 
words, is — 

Subtract the first term from the last, and divide the remain- 
der by the 7iumber of terms less one ; the quotient will be the com- 
mon difference. 

Multiplying both members of the equation (121.) by 
{n — i) and dividing by d, we obtain — 



434 ALGEBRA. 

I — a 



n — \ — 



Transferring i to the second member, we have — 

I — a , . 

n^—^+i', (122.) 

which is a rule for finding the number of terms, and which, 
in words, is — 

Divide the differejtce between the first and last terms by the 
common difference ; to the quotient add unity, a7td the sum will 
be the number of terms. 

Thus it has been shown, in equations (119,) (120), (121), 
and (122), that when, of the four quantities in arithmetical 
progression, any three are given, the fourth may be found. 

The sum of the terms of an arithmetical progression may 
be ascertained by adding them ; but it may also be had by a 
shorter process. If the terms are written in order in a hori- 
zontal line, and then repeated in another horizontal line be- 
neath the first, but in reversed order, as follows : 

i» 3, 5, 7, 9. II. 13, 15. 

15, 13, II, 9, 7, 5, 3, I, 

16, 16, 16, 16, 16, 16, 16, 16, 

and the vertical columns added, the sums will be equal. In 
this case the sum of each vertical couple is 16, and there are 
8 couples; hence the sum of these 8 couples is 8 x 16 = 128. 
And in general the sum will be the product of one of the 
couples into the number of couples. It w^ll be observed 
that the first couple contains the first and last terms, i and 
15 ; therefore the sum of the double series is equal to the 
product of the sum of the first and last terms into the 
number of terms. Or if 5 be put to represent the sum of 
the series, we shall have — 

2 S — (a \ I) n, 

and, dividing both sides by 2 — 

5 = (^ + /)^; (123.) 



GEOMETRICAL TROGRESSION EXPLAINED. 435 

Or, in words : The sum of an arithmetical scries equals the prod- 
uct of the stun of the first aud last terms j into half the number 
of terms. 

4-30. — Geometrical Progression. — A series of numbers, 

such as I, 2, 4, 8, 16, 32, 64, 128, 256, etc., in whicli any one 

of the terms is obtained by multiplying the preceding one 

b}^ a constant quantity, is termed a Geometrical Progression. 

The constant quantity is termed the common Ratio, and is 

equal to any term divided by the preceding one. Thus in 

, 16 8 4 
the above example -5- or - or - = 2, equals the common ra- 

04^ 

tio of the above series. In the series, r, 3, 9, 27, etc., we 
have for the ratio — 

_27 ^ 9 ^ 3 ^ 
931 

which is the common ratio of this series. 
A geometrical series may be put thus : 

Terms: i, 2, 3, 4; 

Progress. : i, i x 3, i x 3 x 3, i x 3 x 3 x 3 ; 

or thus — 

Terms: i, 2, 3, 4: 

Progress.: i, i x 3, i x 3^ i x 3^ 

in which the common ratio, in this case 3, appears in each 
term and with an exponent which is equal to the number of 
terms preceding that in which it occupies a place. 

If the first term be represented by a and the common ra- 
tio by r, then the following will represent any geometrical 
progression — 

a, ar, ar-, ar^, ar'^, etc. (124.) 

For example, let a = 2 and r == 4 ; then the progression 
will be — 

2, 8, 32, 128, 512, etc. 



43^ ALGEBRA. 

li r = unity, then when a = 2 the progression becomes — 

2, 2, 2, 2, 2, etc. 

If r be less than unity, then the progression will be a de- 
creasing one. 

For example, let a = 2 and r =z ^. Then we have for 
the progression — 

I I I I 
' ' 2' 4' 8' 16' ^^^• 

If the number of terms be represented by ?/, and the last by 
/, then the last term w4ll be — 

/ = ar^-\ (125.) 

For example, let 71 equal 6, then the progression will be — 

Terms: i, 2, 3, 4, 5, 6; 

Progress.: a, ar, ar^% ar^, ar"^, ar^\ 

in which the exponent of the last term equals n — \ — 

6-1 = 5. 

If wS be put for the sum of a geometrical progression, we 
will have — 

Multiply each member by r, then — 

Sr =. ar -^ ar^ + j^ cir^~^ -^r ar^~^ Jr cif^- 

Subtract the upper line from the lower ; then — 

5r= ar + ar- -^ ^ ar^'^ -\. ar^'^ + ar'^^ 

S z^ a -^ ar ^ ar"" -\- +^?'"~^ + <^r""' 

Sr — s — — a "^ '^ ^ + ar'\ 

Sr — s = — a + ar'\ 

S {r — i)= — a + (Z r^ = ar^ — a, 
ar^ — a • 

o — • 

r — \ 



GEOMETRICAL PROGRESSION CONTINUED. 437 

The last term (equation (125.)) equals I— ar^-\ and since 
ar"" = r X ar"^' '' = r I, therefore — 

5=^f. (.26.) 

Thus, to find the sum of a geometrical progression : Multi- 
ply the last term by the ratio ; from the product deduct the first 
term, and divide the remainder by the ratio less unity. 

For example, the sum of the geometrical progression — 

5 = I + 3 + 9 + 27 + 81 + 243 + 729 = 1093 

by actual addition. 

To obtain it by the above rule — 



rl — a 3 X 720 — I 

S — — — - — ^^ = 1003, 

r — I 3 — 1 

the correct result. 

If there be a decreasing geometrical progression, as i, ^, 
I, ^j, etc., in which the ratio equals ^, the sum will be — 

^ - I I I I ■ r ' 

5=1 » T + ~ + ^ + "gY' ^tc, to mhnity. 
Multiply this by 3, and subtract the first from the last — 



25 = 3+1+- + - + — + — - + to infinity. 
3927^ 81 ^ 



I I I 

3"^ 9 "^ 27 
2 5 = 3, or 5 = 1^. 



S = i+- + -_|_-^ + --^ + to infinity. 



In a decreasing progression let r, the common ratio, be 
represented by - {b less than e), and the first term by a, then 
'he sum will be — 

b b^ b' . ^ . 

S=-a + a-^a-^-\-a-^+, etc., to mhnity. 



43^ ALGEBRA. 

Multiply this by -, and subtract the product from the 



above — 



^b b b' F . , . 

0- — « — -i-^— +<3:^ 4- etc., to innnity. 

b b^ b' . ^ . 

S ^ a ^ a — -f-rt^-^ + t^— + to mnnity. 



b b b' b' 

— z=i a - 
c c 



S — ^=^ a — \- a —-\- a -^ + \.o infinity. 



C 



Or— 6" (i - -) -: ^ 



a 



■5 = r-7. (127.) 



I -b_ 

C 



For example, let the first term of a geometrical progression 
equal 2, and the ratio equal J, then the sum will be — 



5 __«__,? 



From this, therefore, we have this rule for the sum of an in- 
finite geometrical progression, namely : Divide the first term 
by miity less the ratio. 



SECTION X.— POLYGONS. 

431. — Relation of Sum and Bfifference of Two Liines. — 

Let AB and CD {Fig. 285) be two given lines; make EH 



B 
D 



E 1— 1—1 G 

H J F 

Fig. 285. 

equal to AB, and HG equal to CD\ then £ (S^ equals the 

sum of the two lines. 

Make FG equal to A B, which is equal to EH, 
Bisect EGixi y\ then, also, J bisects HF\ for — 

E7= JG, 
and — 

EH=FG, 

Subtract the latter from the former ; then — 

Ey- EH=yG-FG\ 
but— 

Ey-EH = Hy, 

and — 

yG-~FG = yF] 
therefore — 

Hy-JF. 

Now, E y IS half the sum of the two lines, and Hy is half 
the difference ; and — 

Ey-Hy = EH=AB. 

Or : Ha// the sum of two quantities, minus half their dif- 
ference, equals the smaller of the two quantities. 



440 



POLYGONS. 



Let the shorter line be designated by ^, and the longer 
by b ; then the proposition is expressed by — 



a — 



a+ b 



(128.) 



We also have EJ + JF— EF— CD\ or, half the sum 
of two quantities, plus half their differ ence, equals the larger 
quantity, 

432. — Perpendicular, in Triangle of Known Sides. — 

Let ABC (Fig. 286) be the given triangle, and CE a perpen- 
dicular let fall upon A B, the base. Let the several lines of 




the figure be represented by the symbols a, by c, d, g, and /, 
as shown. Then, since A EC and B E C are right-angled 
triangles, we have {Art. 416) the following two equations, 
and, by subtracting one from the other, the third — 

f^ + d^=a\ 



f^-^'^a'-b'. 
Then {Art. 414), by substitution, we have — 

By division we obtain — 

(a + 6) {a — b) 



f-g^ 



f + g 



RULE FOR TRIGONS. 44I 

According to Art. 431, equation (128.), we have — 

^2 2 ' 

In this expression let the value of / — ^, as above, be 
substituted, then we will have — 

^^/+^ _ {a + b ){a- h) 

Multiply the first fraction by {f + g), then join the two 
fractions, when we will have — 

, ^ {f + gf-{a-\-b){a- b) 

The lines / and g, in the figure, together equal the line 
c ; therefore, by substitution — 

c" — (a + b^ (a — b) , . 

g= —YT "• ^'^^'^ 

This is the value of the line g. 

It may be expressed in words, thus: The shorter of the 
two parts into which the base of a triangle is divided by a 
perpendicular let fall from the apex upon the base, equals the 
quotient arising from a division by twice the base, of the differ- 
ence betiveen the square of the base and the product of the sum 
and difference of the two inclined lines. 

As an example to show the application of this rule, let 
^ = 9, ^ z= 6, and c = \2\ then equation (129.) becomes — 

^ i2--(9 + 6)(9- 6) 
^ 2 X 12 



^^ 144- 15 X 3 
24 

"^ 24 ^8 



442 POLYGONS. 

Now, to obtain the length of <f, the perpendicular, by the 
figure, we have — 

and, extracting the square root — 



d=Vb-'-g\ (130.) 

or, in words : The altitude of a triangle equals the square root 
of the difference of the squares of one of the inclined sides and 
its base. 

As an example, take the same dimensions as before, then 
equation (130.) becomes — 



d=^%'-A¥' 

The square of 6 = 36- 

" '' 4^=: 17.015625 

6^-4F- 18-984375, 
the square root of which is 4-44234; therefore- 



^=^6^-41^ = 4.44234. 

This may be tested by applying the rule to the other in- 
clined side and its base — 



Then, 



c = 


12 


g = 


4i 


f= 


7l- 


d='^g 


'-TV, 


9' = 


81 • 


7'i^ 


62-015625 


-?¥ = 


18-984375- 



TRIGON — RADIUS OF CIRCLES. 



443 



The same result as before, producing for its square root 
the same, 4-zj4234, the value of d', therefore — 



d=^\ 



.7x2 

/8 



4.44234. 



4-33. — Trigon : I&adiu§ of Circwmscribed aaid Inscribed 
Circles: Area. — \.^t A B C (/^2> 287) be a given trigon or 
triangle with its circumscribed and inscribed circles. Draw 
the lines A D F, D B and D C. 

The three triangles, A B D, A CD, :ind B D C, have their 
apexes converging at D, and form there the three angles, 
A D B, ADC, and B D C. These three angles together form 




four right angles {Art. 335), and each of them, therefore, 
equals f of a right angle. 

The angles of the triangle BDC together equal two 
right angles {Art. 345). As above, the angle BDC equals | 
of a right angle, hence 2 — ^ = ^^^ = f of a right angle, 
equals the sum of the two remaining angles at B and C 
The triangle BDC is isoceles {Art. 338); for the two sides 
B D and D C, being radii, are equal ; therefore the two angles 
at the base B and C are equal, and as their sum, as above, 
equals f of a right angle, therefore each angle equals -J of a 
right angle. Draw the two lines FC and FB. Nov/, be- 



444 POLYGONS. 

cause D C and D F are radii, they are equal, hence DFC is 
an isoceles triangle. 

It was before shown that the angle B D C equals f of a 
right angle ; now, since the diameter A F bisects the chord 
B C, the angles B D E and E D C are equal, and each equals 
the half of the angle B D C; or, -J of f of a right angle equals 
f of a right angle. Deducting this from two right angles 
(the sum of the three angles of the triangle), or 2 — f = 
ji nr |- of a right angle equals the sum of the angles at F and 
C\ hence each equals the half of f, or | of a right angle; 
therefore the triangle D FC is, equilateral. The triangles 
DBF and DFC are equal. The angles B D C and B FC are 
equal; the line B C is perpendicular to DF and bisects it, 
making DE and EF equal; hence DE equals half DF, or 
DB, radii of the circumscribing circle. Therefore, putting 
R to represent B D, the radius of the circumscribing circle, 
and I? = B C, a side of the triangle A B C, by Art. 416, we 
have — 

BD'' = BE'+ DE\ 



^■=(D"-(iy 



Transferring and reducing — 



R 



(f)=ej 



4 4 ' 

V 4/ 4 

^R' = -b\ 
4 4 

3 4 3 3 

.^±. 

Or, The Radius of the circumscribing circle of a regular trigon 
or equilateral triangle, equals a side of the triangle divided by 
the square root of 3. 



AREA OF EQUILATERAL TRIANGLE. • 445 

By reference to Fig. 287 it will be observed, as was 
above shown, that D E — E F= = ; or, D E, the ra- 
dius of the inscribed circle, equals half the radius of the 
circumscribed circle; or, again, dividing equation (131.) by 
2, we have — 

R b 



2 2 ^3 ' 

and, putting r for the radius of the inscribed circle, we 
have — 

(132.) 



2 ^3 

Or: TJie radhis oi the inscribed circle of a regular trigofi equals 
the half of a side of the trigon divided by the square root of 3. 
.To obtain the area of a trigon or equilateral triangle ; we 
have {Art. 408) the area of a parallelogram by multiplying 
its base into its height ; and {Arts. 341 and 342) the area of a 
triangle is equal to half that of a parallelogram of equal base 
and height, therefore, the area of the triangle B D 0{Fig. 287) 
is obtained by multiplying B C, the base, into the half of 
ED, its height. Or, when iV^is put for the area — 

N= BCx^^, 

2 

or- N=bx^; 

4 
substituting for R its value (131.) — 

N= bx 

N = 



4 V3 
^2 



4V3 



This is the area of the triangle BBC. 

The triangle A B C is compounded of three equal tri- 
angles, one of which is the tr'mngleBDC; therefore the 
area of the triangle ABC equals three times the area of the 
triangle BBC; or, when A represents the area — 



446 



POLYGONS. 



3^ 

4 Vi 



(•33-) 



Or: The area of a regular trigon or equilateral triangle 
equals tJiree fourtJis of the square of a side of the triangle di- 
vided by the sqiiare root of 3. 



434. — Tetragon ; Radius of Circumscribed and In- 
scribed Circles: Area. — Let AB CD {Fig. 288) be a given 
tetragon or square, with its circumscribed and inscribed 




Fig. 288. 

circles, of which ^ ^ is the radius of the former and E F ihdit 
of the latter. The point F bisects A B, the side of the 
square. A /^equals F F and equals half A B, a side of the 
square. Putting R for the radius of the circumscribed 
circle and b for A B, we have {Art. 416)— 



AE^^ = AF'^-\-EF\ 



-•=(4)MI)'-(I)' 



2 /; 



b- 



R=: 



v 



(134.) 



Or: The radius of the circutuscribed c\yc\q of a regular tetra- 
gon equals a side of the square divided by the square root of 2. 



SIDE AND AREA OF HEXAGON. 



447 



By referring to the figure it will be seen that the radius 
of the inscribed circle equals half a side of the square — 



(•35-) 



The area of the square equals the square of a side — 

A^b\ (136.) 



435. — Hexagon: Radius of Circumscribed and In- 
scribed Circles : Area.— Let ABODE F{Fig. 289) be an equi- 
lateral hexagon with its circumscribed and inscribed circles, 
of which EG xs the radius of the former, and 6^ //that of 
the latter. The three hues, AD, B E, ^nd C/^, divide the 




Fig. 289. 

hexagon into six equal triangles w^ith their apexes converg- 
ing at G. The six angles thus formed at G are equal, and 
since their sum about the point G amounts to four right 
angles {Art. 335), therefore each angle equals -f or | of a 
right angle. The sides of the six triangles radiating from G 
are the radii of the circle, hence they are equal ; therefore, 
each of the triangles is isosceles {Art. 338), having equal angles 
at the base. In the triangle EGD, the sum of the three 
angles being equal to two right angles {Art. 345), and the 
angle at 6^ being, as above shown, equal to f of a right angle, 
therefore the sum of the two angles at E and D equals 
2 — f = f of a right angle ; and, since they equal each other. 



448 POLYGONS. 

therefore each equals f of a right angle and equals the angle 
at G ; therefore E G D is an equilateral triangle. Hence 
E D^2i side of a Jiexagon, equals E G, the radius of the circum- 
scribing circle — 

R=b. (137.) 

As to the radius of the inscribed circle, represented by G Hy 
a perpendicular from the centre upon ED, the base; the 
point H bisects E D. Therefore, E H equals half of a side 
of the hexagon, equals half the radius of the circumscribing 
circle. Let R = this radius, and r the radius of the inscribed 
circle, while d = a, side of the hexagon ; then we have (Arts. 
353 and 416)— 

GII^ = EG'-EH\ 



'-=R 



(4)' 







^2 ._ 




Now, 


i? = 


= d, therefore- 


- 






r=/i^ 


^3 



(138.) 

Or : The radius of the inscribed circle of a regular hexagon 
equals the half of a side of the hexagon, multiplied by the 
square root of 3. 

As to the area of the hexagon, it will be observed that the 
six triangles, A B G, B G C, etc., converging at G, the centre, 
are together equal to the area of the hexagon. The area of 
E G D, one of these triangles, is equal to the product of E D, 
the base, into the half of G H, the perpendicular; or, when 
N is put to equal the area — 

N — EDy. , 



SIDE AND AREA OF OCTAGON. 



449 



2 



and, since r, as above, equals ^ 3 — , 

2 






4 

This is the area of one of the six equal triangles ; therefore, 
when A is put to represent the area of the hexagon, we have — 

^ = 6x |/"3 — , 



^3 



3^ 



(I39-) 



Or: The area of a regular hexagon equals three halves of the 
square of a side multiplied by the square root of '^^ 




Fig. 2go. 

4-36b — Octagon : Radius of Circiimscribed and In- 
scribed Circles: Area. — Let C E D B F {Fig. 290) represent a 
quarter of a regular octagon, in which i^is the centre, ED 
a side, and C E and D B each half a side, while C F and 
B F are radii of the inscribed circle, and EF and DF are 
radii of the circumscribed circle. 



450 POLYGONS. 

Let R represent the latter, and r the former ; also let I? 
represent £ B, one of the sides, and 7i be put for A D, and 
for A E. Then we have — 

AD^DB^CF. 



b 
or — n=^r — — • 

2 

Since ^ Z^^is a right-angled triangle {Art. 416), we have- 
AE' + AD' = ED\ 

n'+ii' = b\ 

2 7l' = b\ 

n — — 
2 



n^^\ 



Placing the value of n, equal to the value before found, 
we have — 



-? = i/^, 



r 



= '/^*j + ^=il*!+^, 



- A ^_ 



V 



o 



V2 2 



(7; * -;)'■ 



This coefficient may be reduced by multiplying the first 
fraction bv 1/2, thus — 



V2 "\/~2 ~ 2 



RULES FOR OCTAGON. 45 1 

therefore — 

V~2 



\ 2 2 / 2 



r^{V2 + i)— . , (140.) 

Or: The radius of the inscribed circle of a regular octagon 
equals half a side of the octagon multiplied by the sum of unity 
plus the square root of 2. In regard to the radius of the cir- 
cumscribed circle, by Art. 416 we have — 

In this expression substituting for r\ its value as above, we 
have — 






The square of the coefficient ( -/2 + i ) by Art. 412 equals 
2 + 24/2+1 = 2 -/a +3, then — 

R 



^ = [(2 4/3 + 3) + I] (4-)'- 
ie^ = (2Vj+4)(Ay. 

i?=|/2l/2' + 4-^. (141.) 



Or: The radius oi the circtcmscribed c\yc\q oi a regular d?^/^^'^?;^ 
equals half a side of the octagon multiplied by the square root of 
the sum of twice the square root of 2 plus 4. 

In regard to the area of the octagon, the figure shows 
that one eighth of it is contained in the triangle D E F. 



452 POLYGONS. 

The area oi D E F^ putting it equal to N, is- 

^ ^ BF 
N — E D ^ , 



2 



^^^.(^2+,) 2 



X 



2 

N^ {V2+ I)--. 

4 

This is the area of one eighth of the octagon ; the whole 
area, therefore, is — 

4 

A =: ( 4/2"+ l) 2 ^\ (142.) 

Or: The area of a regular octagon equals twice the square of a 
side, multiplied by the sum of the square root of 2 added to unity. 
When a side of the enclosing square, or diameter of the 
inscribed circle, is given, a side of the octagon may be found ; 
for from equation (140.), multiplying by two, we have — 

2 r — {\^2 -\- \) b. 
Dividing by S^ 2 ^ i, gives — 

T 2r , . 

^ = -— -— . (143.) 

r 2 + I 

The numerator, 2 r, equals the diameter of the inscribed 
circle, or a side of the enclosing square ; therefore : 

The side of a regular octagon, equals a side of the enclosing 
square divided by the sum of the square root of 2 added to unity, 

437,— Dodecagon : Radiu§ of Circuniseribed and In- 
scribed Circles: Area. — Let A B C {Fig. 291) be an equilat- 



SIDE AND AREA OF DODECAGON. 



453 



eral triangle. Bisect A B in F; draw C F D ; with radius A C 
describe the arc A D B. Join A and D, also D and B ; bisect 
A F> in E ; with the radius £ C describe the arc E G, Then 
A D and D B are sides of a regular dodecagon, or twelve- 
sided polygon ; of which A C, D C, and B C are radii of the 
circumscribing circle, while ^ 6" is a radius of the inscribed 
circle. 

The line A B is the side of a regular hexagon (Arl. 435). 
Putting R equal to A C the radius of the circumscribing cir- 
cle \ r, =^ E C, the radius of the inscribed circle ; i^, = A D, 3. 
side of the dodecagon, and ;/ — B> F. Then comparing the 




Fig. 291. 

homologous triangles, ABE and A EC (the angle ADF 
equals the angle E A C, and the angles BE A and A E C are 
right angles); therefore, the two remaining angles DAE 
and A C E must be equal, and the two triangles homologous 
{Art. 345). Thus we have — 

DF '.DA :: AE : A C, 



n \ b \\~ \ R, 

2 



R 



b^ 

2 n 



454 POLYGONS. 

In Art. 435 it was shown that FC {Fig. 291), or G H oi 

_ b 
Fig. 289, the radius of the inscribed hexagon, equals 1^ 3 ~, 

and in which its <^ = i? ; Fc = |/7 — . 

Now ( Ftg. 291) — 

DF= DC - FC, 
or — 

Substituting this value of n^ in the above expression, we 
have — 

^^ ^1 

2 i? (I - i V 3)* 

Multiplying by R and reducing, we have — 



R^^ = 



V7 



R = '/— ^^ ^. (144.) 

2-1/3 

Or : The radius of the circumscribed circle of a regular dodec- 
agon, equals a side of the dodecagon multiplied by the square 
root of a fraction, having unity for its numerator and for its 
denominator 2 mimis the square root of 3. 

Comparing the same triangles, as above, we have — 

FD \ FA :: EA \ E C, 

or — 

R b 

n : — : : — '. r, 

2 2 

Rb Rb 

r = 



An 4^(1 -it' 3)* 

r = ~.^. (145.) 

4-21/3 



RULE FOR DODECAGON. 455 

Or : The radius of the inscribed circle of a regular dodecagon 
equals a side of the dodecagon divided by the difference between 
4 and the square root of 3. 

The area of a dodecagon is equal to twelve times the area 
of the triangle ADC {Fig. 291). The area of this triangle is 
equal to half the base by its perpendicular ; or, A E x EC\ 
or — 

b 

2 ""' 

or, where N equals the area — 

N=^br. 

Or, for the area of the whole dodecagon— 

12 N — 6 br, 
A =6br. 

Substituting for r its value as above, we have — 

A = --^-- b^. (146.) 

4 — 24/3 

Or : The area of a regular dodecagon equals the square of a 
side of the dodecagon, multiplied by a fraction having 6 for its 
numerator, and for its denominator, 4 minus twice the square 
root of T^. 

438a — Hecadecagon : !&ad§ii§ of 4 ircumscribed and In- 
scribed Circles: Area. — Let A BCD {Fig. 292) be a square 
enclosing a quarter of a regular octagon C E F B^ E F being 
one of its sides, and C E and F B each half a side, while ED 
is the radius of the circumscribed circle, and J D the radius 
of the inscribed circle of the octagon. Draw the diagonal 
A D ; with D F ior radius, describe the circumscribed circle 
EGF', join G Avith F and with E; then E G and G F W\\\ 
each be a side of a regular hecadecagon, or polygon of six- 
teen sides. 

An expression for F D, the radius of the circumscribed 



456 



POLYGONS. 



circle, may be obtained thus: Putting FD = R; H D =r r 
GF = b; Gy = n\ and JF = - {Art. 416), we have— 




C D 

Fig. 292. 

Comparing the two homologous (ArL 361) triangles, G^F 
and FH D {Art. 374), we have — 

GJ '. GF '.: HF : FD, 

n \ b -.'. '- '. R, 



n = 



2R' 



n" = 



4R'- 



Putting this value of ;/' in an equation against the former 
value, we have — 

^■"•-& 

In Art. 436, the value of F D, as the radius of the cir- 
cumscribed circle of a regular octagon, is given in equation 
(141.) as— 



R=^2V2+4-y 

2 



SIDE AND AREA OF HECADECAGON. 457 

in which b represents a side of the octagon, or E Fy for 
which we have put s. Substituting s for b and putting the 
numerical coefficient under the radical, equal to B^ we 
have — 

Squaring each member gives — 

From which, by transposition, we have — 

Substituting in the above expression for (-- ) , this value 

of it, gives — 

-^-^b^-^\ 
AR' B 

Transposmg, we have— 

^--^^'^b\ 
4R' B 

Multiplying the first term by B, and the second by 4 7^', 
we have — 

4BR' 4BR' 

Bb'-^^ R^ _ 
4BR' ~ ' 

Bb' +^R'^^BR'b\ 

Transposing, we have — 

^R'-^BR'b'= -Bb\ 



458 POLYGONS. 

To complete the square {ArL 428) we proceed thus — 
R'-BR'y-= -\Bb\ 
R'-BR"'h' + {l-Bby=^ {IBby -lBb\ 
Taking the square root, we have — 



R'-^Bb'= V^B'b'-lBb 



R'= ViB'b'-iBb'+ iBb\ 



R' = bWiB'-iB + iBb\ 



R''=b\V-lB-'-iB + iB), 



R'=b"'{V\B{B - i) + ^B). 
Restoring B to its value, 2 4/2 + 4 as above, we have — 



\B = ^^\^+ I, 

^—1 = 21/24-3; 



multiply these — 

2 + 2 V2, 



tV 



3 + t ^ 2 



j^(i?- I) = 5 + i-^2, 

^B= V2 + 2. 
Therefore— 

R'-^b'{^S ■\-lV~2+V2 + 2), 



R=b\/ ^\ + I V2 + V~2 + 2. (147.) 



RULES FOR HECADECAGON. 459 

Or: The radms of the circumscribed circle of a regular 
Jiccadecagon equals a side of the Jiecadecagon multiplied by the 
square root of the sum of two quantities, one of which is the 
square root of 2 added to 2, and the other is the square root of 
the sum of seven halves of the square root of 2 added to 5. 

To obtain the radius of the inscribed circle we have {Fig. 
292)— 

HD' ^ FD'- HF\ 

Substituting for R"" its value as above, we have — 



r^ = b^[_{ViB{B-i)+iB)-{iYl 



The coefficient of b is the same as in the case above, ex- 
cept the — \ ; therefore its numerical value will be i less, 
or — 



^ V ^'5 + i- 1/2 + 1/2 + if. (148). 



Or: The radius of the inscribed circle ot a regular Jiecadeca- 
gon equals a side of the Jiecadecagon multiplied by tJie square root 
of two quantities^ one of ivJiicJi is tJie square root of 2 added to 
if, and tJie otJier is the square root of the sum of seven Jialves of 
tJie square root of 2 added to 5 . 

To obtain the area of the hecadecagon it will be observed 
that the area of the triangle G FD {Fig. 292) equals HD x 
H F, and 'that this is the -^-^ part of the polygon ; we there- 
fore have — 

A = i6HDxHF, 
A = i6r-= Srb. 



460 POLYGONS. 

The value of r is shown in (148.) ; therefore we have — 



A 



8^ 



j/ ^5 + -1- 1^2 + \'2 + if. (149.) 



Or: The area oi a regular Jiecadccagon equals eight times 
the square of its side^ multiplied by the square root of two quan- 
tities, one of which is the square root of 2 added to if, and the 
other is the square root of the stem of seven halves of the square 
root of 2 added to 5. 

439.— Polygons ^ Radius of Circumscribed and Inscribed 
Circles : Area. — In Arts. 433 to 438 the relation of the radii 
to a side in a trigon, tetragon, hexagon, octagon, dodeca- 
gon and hecadecagon have been shown by methods based 




upon geometrical proportions. This relation in polygons 
of seven, nine, ten, eleven, thirteen, fourteen and fifteen 
sides, cannot be so readily shown by geometry, but can be 
easily obtained by trigonometry — as also said relation of the 
parts in a regular polygon of any number of sides. The na- 
ture of trigonometrical tables is discussed in Arts, 473 and 
474. So much as is required for the present purpose will 
here be stated. 

Let ABC {Fig. 293) represent one of the triangles into 
which any polygon may be divided, in which B C =^ b = 71 
side of the polygon; A C — R = the radius of the circum- 
scribed circle ; and A D — r = the radius of the inscribed 
circle. 



GENERA.L RULES FOR POLYGONS. 461 

Make ^6^ equal unity ; on 6^ as a centre describe the arc 
E F\ draw F H and E G perpendicular to B C, or parallel to 
A D ; then for the uses of trigonometry E G is called the 
tangent of <r, or of the angle A C B, and F H \s the sine, and 
H C the cosine of the same angle. 

These trigonometrical quantities for angles varying from 
zero up to ninety degrees have been computed and are to be 
found in trigonometrical tables. 

Referring now to Fig. 293 we have — 

HC '. EC :\ DC '. AC, 



b 
COS. c \ \ \\ - \ R 

2 



b 



R ^ . (150.) 

2 COS. C \ J / 



Again — 

E C : E G : : D C : A n, 

b 
I : tan. e : : — : r, 

r = - tan. r. (151.) 

These two equations give the required radii of the cir- 
cumscribed and inscribed circles. They may be stated thus : 

The radius of the circumscribed circle oi any regular poly- 
gon equals a side of the polygon divided by twice the cosine of 
the angle formed by a side of the polygon and a radius from one 
end of the side. 

The radius of the inscribed circle of any regular polygon 
equals half of a side of the polygon multiplied by the tangent of 
the angle formed by a side of the polygon and a radius from one 
end of the side. 

The area of a polygon equals the area of the triangle 
ABC {Fig. 293), (of which B C is one side of the polygon 
and A is the centre), multiplied by the number of sides in 
the polygon ; or, if;/ be put to represent the number of the 
sides and A the area, then we have — 



4^2 POLYGONS. 

A = Bit, 

in which B equals the area of the triangle. The area of A 
B C {Fig. 293) is equal X.o AD x B D, or — 

2 

For r substituting its value, as in equation (151.), we have — 

B ^ - tan. c- — -b" tan. c, 

2 2 4 • 

Therefore, by substitution — 

A — - b'^ n tan. c. (152.) 

Or : The area of a regular polygon equals the square of a 
side of the polygon, multiplied by one fourth of the number of 
its sides, and by the tangent of the angle formed by a side of the 
polygon, and a radius from one end of the sides. 

440. — PoBygons : Til eir Angles. — Let a line be drawn 
from each angle of a regular polygon to its centre, then 
these lines form with each other angles at the centre, which 
taken together amount to four right angles, or to 360 de- 
grees {Arts. 327, 335). 

If this 360 degrees be divided by the number of the sides 
of the polygon, the quotient will equal the angle at the cen- 
tre of the polygon, of each triangle formed by a side and two 
radii drawn from the ends of the side. For example: if 
ABC {Fig. 293) be one of the triangles referred to, having 
B C one of the sides of the polygon and the point A the cen- 
tre of the polygon, then the angle B A ^will be equal to 360 
degrees divided by the number of the sides of the polygon. 
If the polygon has six sides, then the angle B A (7 will contain 

—p- = 60 degrees ; or if there be 10 sides, then the angle at 

-360 
^, the centre, will contain = 36 degrees. The angle 



SIDE AND AREA OF PENTAGON. • 463 

BAD equals half the angle BAC^ or, when 11 equals the 
number of sides, the angle B A C equals — 

360 
n 

B A C 
and the triangle B A D ^= , equals — 

2 ji 

Now the angles B A D + D B A equal one right angle 
{Art. 346), or 90 degrees. Hence the angle DBA — go° — 
BAD,or the angle c equals — 

^ 2 71 

For example, if u equal 6, or the polygon have six sides, 
then — 

"° = 90°- ^=90-30=60°. 

Therefore, the a;i£-/e c, contained in equations (150.), (151.)' 
and (152.), equals 90 degrees, less the quotient derived front a di- 
vision of 360 by twice the Jiicinber of sides to the polygon. 

4-4-(. — Peiatag^on: lladiu§ of tlie Circwmscritoed anal In- 
scribed Circles: Area. — The rules for polygons developed 
in the two former articles will here be exemplified in their 
application to the case of a regular pentagon, or polygon of 
five sides. 

To obtain the angle t° (i53-)> we have n — 5, and — 

.° = 90° -^=90-36 = 54°. 

For the radius of the circumscribed circle, we have 

(150.)- 

b 



R = 



2 COS. c 



464 POLYGONS. 



2 COS. 54 

ie = ^ ^ — ^. 

2 COS. 54 

Using a table of logarithmic sines and tangents {Art. 427), 
we have — 

Log. 2 = 0-3010300 
Cos. 54° = 9-7692187 

Their sum = 0-0702487 — subtracted from 
Log. I = o- 0000000 

0-85065 =9-9297513 

Therefore — 

i?= 0-85065 <^. 

Or : The radius of the circumscribed circle of a regular pcnta- 
gon equals a side of the pentagon multiplied by the decimal o - 8 5065 . 

For the radius of the inscribed circle, we have (151.) — 

b 
r = — tan. c, 

2 ' 

, tan. 54° 
r — b 

2 

For this we have — 

Log. tan. 54° = 0-1387390 
Log. 2 = 0-3010300 

0-68819 = 9-8377090. 
Therefore — 



19 b. 

Or: The radius oi i\\Q inscribed circle of a regular /^^^/^^<^;/ 
equals a side of the pentagon multiplied by the decimal 0'62)^ig. 
For the area we have (152.) — 

A =1 b'^n tan. r, 

^ = i X 5 tan, 54° ^', 

^=f tan. 54° b\ 



TABLE FOR REGULAR POLYGONS. 



465 



For this we have — 

Log. 5. = 0-6989700 
Log. tan. 54° =-- 0-1387390 

0-8377090 
Log. 4 = 0-6020600 

I -72048 11=0-2356490 
A — 1. 72048 1? \ 



Therefore 



Or: The area of a regu\a.r pentagon equa/s tJie square of its 
side multiplied by i • 72048. 

/|-42- — Polygons: Table of Constant MnUipHers. — ^ To 

obtain expressions for the radii of the circumscribed and in- 
scribed circles, and for the area for polygons of 7, 9, 10, 11, 
13, 14, and 15 sides, a process would be needed such pre- 
cisely as that just shown in the last article for a pentagon, 
except in the value of n and r, which are the only factors 
which require change for each individual case. 

No useful purpose, therefore, can be subserved by ex- 
hibiting the details of the process required for these several 
polygons. The values of the constants required for the 
radii and for the areas of these polygons have been com- 
puted, and the results, together with those for the polygons 
treated in former articles, gathered in the annexed Table of 
Regular Polygons. 

REGULAR POLYGONS. 



Sides. 



3- 
4. 
5. 
6. 

7. 
8. 

9- 
10. 
II. 
12. 
13. 
14. 
15- 
16. 



Trigon 

Tetragon 

Pentagon 

Hexagon i i 



Heptagon 

Octagon 

Nonagon 

Decagon 

Undecagon. . . 
Dodecagon . . 
Tredecagon. . , 
Tetradecagon. 
Pentadecagon 
Hecadecagon. 



57735 

70711 

85065 

00000 

15238 

30656 

46190 

61803 

77473 

93185 

2-08929 

2-24698 

2-40487 

2-56292 



.28868 

• 50000 
•68819 

• 86603 

1 03826 
I -20711 
1-37374 
1-53884 
1-70284 
I • 86603 
2-02858 

2 - 1 9064 
2-35231 
2-51367 



•43301 
I • 00000 
1-72048 
2-59808 
3-63391 
4.82843 
6-18182 
7-69421 

9-36564 
11-19615 

13-18577 

15-33451 
17-64236 
20-10936 



466 POLYGONS. 

In this table R represents the radius of the circumscribed 
circle ; r the radius of the inscribed circle ; b one of the 
sides, and A the area of the polygon. By the aid of the 
constants of this table, R, the radius of the circumscribed 
circle of any of the polygons named, may be found when a 
side of the polygon is given. For this purpose, putting m 
for any constant of the table, we have — 

R — bin. (154-) 

As an example : let it be required to find R, for a penta- 
gon having each side equal to 5 feet ; then the above expres- 
sion becomes — 

i? = 5 X 0-85065, 
R = 4-25325. 

The radius will be 4 feet 3 inches and a small fraction. 
In like manner the radius of the inscribed circle will be — 

r = bm; (155.) 

and for a pentagon with sides of 5 feet, we have — 

r = 5 X 0-68819, 
r =3.44095. 

Or, the radius of the inscribed circle will be 3 ft. -^q and a 
small fraction. Or, multiplying the decimal by 12, 3 ft. 5 in. 
j^-^Q and a small fraction. 

The area of any polygon of the table may be obtained 
by this expression — 

A = b'7;i; (156.) 

and, applying this to the pentagon as before, we have — 

^ = 5^x1- 72048, 
A = 43-012. 



EXPLANATION OF THE TABLE. 467 

Or, the area of a pentagon having its sides equal to 5 feet, 
is 43 feet and yi|-g- of a foot. 

By the constants of the table a side of any of its poly- 
gons may be found, when either of the radii, or the area, 
are known. 

When R is known, we have — 

" = -■ ('57.) 

When r is known, we have — 

b = --. (158.) 

in 
When the area is known, we have — 



m 



(159.) 



SECTION XL— THE CIRCLE. 

443. — Circle§: Diameter and Perpendicular: mean 
Proportional. — Let ABC {Fig. 294) be a semicircle. From 
C, any point in the curve, draw a line to A and another to 
B\ then ABC will be a right-angled triangle {Art. 352). 
Draw the line CD perpendicular to the diameter A B ; 
then CD will divide the triangle ABC into two triangles. 
A CD and C B D, which are homologous. For, let the 
triangle C B D be revolved on Z> as a centre until its line 
CD shall come to the position i^^ /?, and the hue DB oc- 
cupy the position D F, each in a position at right angles 




to its former position, the point B describing the curve 
B F, and the point C the curve C E, and each forming a 
quadrant or angle of ninety degrees. Since these points 
have revolved ninety degrees, therefore the three lines of 
the triangle 6'^Z>have revolved into a position at right 
angles to that which they before occupied ; hence the line 
FF is at right angles to CB, and (from the fact that A CB is 
a right angle) parallel with A C. Since the triangle E FD 
equals the triangle CB D. and since the lines of E FD are 
parallel respectively to the corresponding lines of A CD, 
therefore the triangles^ CD and CB D are homologous. 

Comparing the lines of these triangles and putting a = 
A B, y = CD, and x — D B, we have — 



RADIUS FROM CHORD AND VERSED SINE. 

DB '. D C-.'. DC : AD, 

X \ y '. \ y \ a — X, 



469 



y'' — X {a — x). 



(160.) 



Or, in a semicircle, 2i perpendicular to the ^/<^;;/^/^t terminated 
by the diameter and the curve is a geometric mean, or mean 
proportional, between the two parts into zvJiicJi the perpendicular 
divides the diameter. 



444. — Circle : Radius from Given Chord and Versed 
Sine.— Let A B {Fig. 295) be a given chord line and CD a 
versed sine. Extend CD to the opposite side of the circle ; 
it will pass through F, the centre. Join A and C, also E and 




B. The line A />, perpendicular to the diameter C E, is 
a mean proportional between the two parts C D ^nd DE 
{Art. 443) ; or, putting a r^ A D, b = C D, 2ind r equal the 
radius FE, we have — 



CD : AD :: AD : DE 
b : a : : a : 2 r — b, 

a' = b{2 r-b\ 

a' = 2 rb-b\ 

a'' ^-b'' — 2rb, 

a^ + b- 

r = ■ ^ — . 

2 b 



(161; 



470 



THE CIRCLE. 



Or : The radius of a circle equals the sum of the squares of half 
the chord a7id the versed sine, divided by twice the versed sine. 

Another expression for the radius may be obtained ; for 
the two triangles C B D and C E B {Fig. 295) are homologous 
(Art. 443) and their corresponding lines in proportion. Put- 
ting/for CB, we have — 



CD 



or — 
or — 
and — 



CB : 


: CB 


: CE, 


:/: 


:/: 2 


^y 


/■- = 


-- 2rv, 




r = 


2V' 





(162.) 



Or : The radius of a circle equals the square of the chord of half 
the arc divided by twice the versed sine. 

4-45. — Circle: Segment from Ordiiiate§. — When the curve 
of a segment of a circle is required for which the radius can- 
not be used, either by reason of its extreme length, or be- 



D^P 




Fig. 296. 

cause the centre of the circle is inaccessible, it is desirable 
to obtain the curve without the use of the radius. This may 
be done by calculating ordinates, a rule for which will now 
be developed. 

Let DC B {Fig. 296) be a right angle, and A DB ^ cir- 
cular arc described from (7 as a centre, with the radius 
BC^CD= CP. Draw PM parallel with DC, and AG 
parallel with C B. Now, in the segment A D G, we have 
given A G,.iis chord, and D E^ its versed sine, and it is re- 



RULE FOR ORDINATES. 47 1 

quired to find an expression by which its ordinates, as P F, 
may be computed. From Art. 416, we have — 



or, putting for these lines their usual symbols — 





^2 __ ^2_ ^2^ 




y^ ^ r^-x\ 


now we have — 






EC^DC-DE, 




EC= FM, 




FM=DC-DE, 




FM^r-b. 


Then we have — 






PF^PM- FM 



or, putting t for P/^and substituting for PJ/and P J/ their 
values as above, we have — 

t = y-{r-b), 

and for y, substituting its value as above, we have — 



t:=^ \/r'^ -x^ -(r-b), (163.) 

Or : The ordinate in the segment equals the square root of the 
difference of the squares of the radius and the abscissa niimis 
the difference of the radius and the versed sine. 

For example : let the chord A G {Fig. 296) in a given case 
equal 20 feet, and the versed sine, b, or the rise D E^ equal 4 
feet ; and let the ordinates be located at every 2 feet along 
the chord line, A G. 

In solving this problem Ave require first to find the radius. 
This is obtained by means of equation (161.) — 

a'^^b^^ 



472 THE CIRCLE. 

For a, half the chord, we have lo feet ; for h, the versed 
sine, we have 4 feet ; and, substituting these values, we 
have — 

10^ + 4- 116 

r — ^— — = 14- 5 

2x4 8 ^ ^ 

The radius equals — 14-5 

The versed sine equals — 4-0 

{r — b)-= 10.5 

The square of 14-5, the radius, equals 210-25. Now we 
have, substituting these values in equation (163.) — 



i— 1/210-25 — ;ir- — 10-5. 

The respective values of x^ as above required, are o, 2, 
4, 6, 8 and 10 Substituting successively for x one of these 
values, we shall have, when — 



X — 


0; 

2 ; 

4; 
6; 
8; 


^ = V 210-25 


— 0-— 10-5 = 4- 


X — 


^ = V 210-25 


— 2- — IO-5 = 3-8614 


X := 


^ — V 210-25 


-4'- IO-5 = 3-4374 


X = 


^ = V 210-25 


— 6-— IO-5 = 2-7004 


X = 


^ = V 210-25 - 


-8^- 10-5 = 1-5934 



X — lu , I — j/210-25 — IO-— 10-5=0-0 

Values for t may be taken at points as numerous as desira- 
ble for accuracy. 

In ordinary cases, however, they need not be nearer than 
in this example. 

After the points are secured, let a flexible piece of wood 
be bent so as to coincide with at least four of the points at a 
time, and then draw the curve against the strip. 

446. — Circle : Relation of Diameter Jo Cireiiinfereiice. 

— In Art. 439 it is shown that the area of a polygon equals 
the radius of the inscribed circle multiplied by half of a 
side of the polygon and by the number of the sides ; or, 



TO FIND THE CIRCUMFERENCE. 473 

b r 

A = r X —n = bn ; or, the area equals half the I'adius by a 

22 ^ 

side into the number of sides ; or, half the radius into the 
periphery of the polygon. Now, if a polygon have ver}^ 
small sides and many of them, its periphery will approxi- 
mate the circumference of the circle inscribed within it; in- 
deed when the number of sides becomes infinite, and conse- 
quently infinitely small, the periphery and circumference 
become equal. Consequently, for the area of the circle, we 
have — 

A = T-., (164.) 

where c represents the circumference. 

By computing the area of a polygon inscribed within a 
given circle, and that of one circumscribed about the circle, 
the area of one will approximate the area of the other in 
proportion as the number of the sides of the polygon are 
increased. 

For example : if polygons of 4 sides be inscribed within 
and circumscribed about a circle, the radius of which is i, 
the areas will be respectively 2 and 4. If the polygons have 
16 sides, the areas are each 3 and a fraction, the fractions 
being unlike; when they have 128 sides the areas are each 
3 • 14 and with unlike fractions ; when the sides are increased 
to 2048, the areas each equal 3-1415 and unlike fractions, 
and when the sides reach 32768 in number the areas are 
equal each to 3-1415926, having Hke decimals to seven 
places. The computations have been continued to 127 
places (Gregory's " Math, for Practical Men "), but for all 
possible uses in building operations seven places will be found 
to be sufficient. From this result we have the diameter in 
proportion to the circumference as i : 3- 141 5926, or as — 

I : 3-i4i59i. 
I : 3-I4I59' 
I : 3-1416. 

Of these proportions, that one may be used which will give 



474 THE CIRCLE. 

a result most nearly approximating the degree of accuracy 
required. For many purposes the last proportion will be 
sufficiently near the truth. 

For ordinary purposes the proportion 7 : 22 is very use- 
ful, and is correct for two places of decimals ; it fails in the 
third place. 

The proportion 113 : 355 is correct to six places of deci- 
mals. 

For the quantity 3-1415926 putting the Greek letter it 
(called py)y and 2r = d for the diameter, we have — 

c — 7t d. (165.) 

To apply this : in a circle of 50 feet diameter, what is 
the circumference ? 

r = 3- 1416 X 50 
c = 157.08//. 

If the more accurate value of tt be used, we have — 

^ = 3' 141 5926 X 50, 
c = 1 57- 07963- 

The difference between the two results is 0-00037, which 
for all ordinary purposes, would be inappreciable. 
By the rule of 7 : 22, we have — 

c == 5ox\2. =: i5;.i42857i, 

an excess over the more accurate result above, of 0-0632271, 
Avhich is about f of an inch. 

Bv the rule of 113 : 355, we have — 

^ = 5oxm= 157-079646. 

This result gives an excess of only o-ooooi6; it is sufficiently 
near for any use required in building. 

From these results we have these rules, namely: To 
obtain the circumference of a circle, multiply its diameter by 



TO FIND THE AREA. 4/5 

22, and divide the product by y ; or, more accurately, multiply 
the diameter by i^^ and divide the produet by ii^; or, by mul- 
tiplication only, multiply the diameter by 3-1416; or, by 
3.14159^; or, by 3-1415926; according to the degree of 
accuracy required. 

And conversely : To obtain the diameter from the cir- 
cumference, multiply the circumference by 7 and divide the 
product by 22', or, multiply by 113 and divide by '^^^\ or, di- 
vide the circumference by 3-1416; or, by 3-i4i59:|-; or, by 
3 -141 5926. 

447- — Circle : LiCnglh of an Arc. — Considering the cir- 
cle divided into 360°, the length of an arc of one degree in 
a circle the diameter of which is unity may be thus found. 

The circumference for 360° is 3 - 141 59265 ; 

3-14159265^^.^^872664625; 
360 

which equals an arc of one degree in a circle having unity 
as its diameter; or, for ordinary use the decimal 0-008727 
or o-oo87i may be taken ; or putting a for the arc and g for 
the number of degrees, we have — 

a — 0-00872665 dg. (166.) 

Wherefore : To obtain the length of an arc of a circle, 

multiply the diameter of the circle by the number of degrees in 
the arc, and by the decimal 0-0087^, or, instead thereof, by 
0-008727. 

448. — Circle : Area. — The area of a circle maybe ob- 
tained in a manner similar to that for the area of polygons 

{Art. 439), in which A^^Bn\ ^5 = r — , or — 

A =z ^ b nr, 

where b equals a side of the polygon and n the number of 
sides ; so that b n equals the perimeter of the polygon. 

Now, if for the perimeter of the polygon there be sub- 



476 



THE CIRCLE. 



stitiited the circumference of the circle, we shall have, put- 
ting for the circumference 3- 1416 d, or, n d {Art. 446) — 

A =^7tdr, 

in which r is the radius. Since 2 r — d, the diameter, and 



r = -, we have- 

2 



And since- 

or — 
Therefore- 



d 
A —\7td — , 

2 



A ^\nd\ 

TT = 3.14159265, 
i^ = 0-78539816, 
^ TT = Q. 7854, nearly. 

A = 0-7854^1 



(167.) 



Or: The area of a circle equals the square of the diameter utiil- 
tiplied by 0-7854. 




D 

Fig. 297. 

As an example, the area of a circle 10 feet in diameter is 
found thus — 

10 X 10 = 100. 

100 X 0-7854 = 78-54 feet. 

449. — Circle: Area of a Sector. — The area oi A B C D 
{Fig. 297), a sector of a circle, is proportionate to that of the 
whole circle. For, as the circumference of the whole circle 
is to its area, so is the arc A B C to the area of A B C D. 



AREA OF SECTOR. 47/ 

The circumference of a circle is (165.) C = 7t d. The area 
of a circle is (167.) A = • 7854 d^\ For the arc ABC put a, 
and for the area of A B C D put s. Then we have from the 
above-named proportion — 

It d : -7854 d'^ \ \ a \ s^ 

.7854 rf' 

K d 

The coefficient 0-7854 is — {Art. 448). 

4 
Therefore, multiplying the fraction by 4, we have — 

n d'- 

S — , a ; 

4 n d 

or — S=^\da — \r a. (168.) 

Wherefore : To obtain the area of a sector of a circle, 
multiply a quarter of the diameter by the lejtgth of the arc. 

Thus: let ^ Z> equal 10; also let A B C = a, equal 12. 
Then the area oi A B C D is — 

5 = ^ X 10 X 12, 

5 = 60. 

The length of the arc may be had by the rule in Art. 44^. 

450. — Circle: Area of a Segment. — In the last article, 
A BCD {Fig. 297) is called the sector of a circle. Of this 
the portion included within A E C B \s> <i segment of a circle. 
The area of this equals the area of the sector minus the area 
of the triangle ADC] or, putting M for the area of the seg- 
ment, S for the area of the sector, and T for the area of the 
triangle, then — 

M-^S- T. 

c 
Putting c for A C {Fig. 297) and Ji for D E, then T = ~ h. 

In the last article, j- = -J r<7, in which a = the length of the 



4/8 



THE CIRCLE. 



arc ABC. Substituting this value of s in the above, we 
have — 



M ^ - r — -h 

-7 '^. 



ar — cJi 



(169.) 



Or : When the length of the arc is known, also that of the 
chord and the perpendicular from the centre of the circle, 
then the area of the segment eqtiah the difference betiveen the 
product of half the arc into the radius^ and half the cJiord into 
its perpendicular to the centre of the circle. 

But ordinarily the length of the arc and of the chord are 
unknown. If in this case the number of degrees contained 
between the two radii, DA, D C, are known, then the area of 



B 





E ^S 




/ 


~^^\y 


/ 


71- 


I 


1/ 

3 




Fig 


298. 





the segment may be found by a rule which will now be de- 
veloped. 

In Fig. 298 (a repetition of Fig. 297) upon Z^ as a centre, 
and with D F — unity for a radius, describe the arc H F. 
Then GF is the sine of the angle C D B, and D G \s the co- 
sine ; and we have — 



DF : GF :: DC : EC. 



or — 



Again- 



or- 



sin : : r \ — ^=. r sin. 

2 



DF : DG :: D C : D E, 
1 : cos : : r \ h =^ r cos. 



RULE FOR AREA OF SEGMENT. 479 

By equation (166.) we have — 

a = 0-00872665 dg, 

in which a is the length of the arc ; g the number of degrees 
contained in the arc ; and d is the diameter of the circle. 
Since d ~ 2 r, therefore — 

a = 0-0174533 rg. 

Putting B for the decimal coefficient, we have — 

a =: Brg. 

The expression (169.), by substitution of values as above, 
becomes — 

M = -r h, 

2 2 

Brg. 
M = r — r sm. x r cos. 

2 

M ^= \ B gr"^ — sin. cos. r^ 

M — r' a Bg — sin. cos.) 

M= r' (0-00872665 ^ — sin. cos.) (i/O.) 

Or : The area of a segment of a circle equals the square of the 
radius into tJie differenee betzveen 0-00872665 times the number 
of degrees contained in the arc of the circle, and the product of 
the sine and cosine of half the arc. 

When the number of degrees subtended by the arc is 
unknown, or tables of sines and cosines are not accessible, 
then the area may be obtained by equation (169.), provided 
the chord and versed sine are known ; but before this equa- 
tion can be used for this purpose, expressions giving their 
values in terms of the chord and versed sine must be ob- 
tained, for a, the arc, r, the radius, and //, the perpendicular 
to the chord from the centre of the circle. 

For the value of the arc we have (from " Penny Cycl.," 
Art. Segment) as a close approximation — 

« = i(8/-f). 



480 THE CIRCLE. 

By equation (162.) we have — 

Then — 

Ji z=z r — V, 
or — 

P 
h^^— - V. 

2 V 

Substituting these values in equation (169.) we have — 



1 



;;-(8/-^)^l-K^l--)]- (^71.) 



This rule is the rule (169.) expanded. 

The written rule for equation (169.) may be used, substi- 
tuting for *' Jialf the arc,'' one sixth of the difference between 
eight times the chord of half the arc and the chord (or -J- of 8 
times A B, Fig. 298, minus A C, the chord). Also substitute 
for '' the radius,'' the square of the chord of half the arc divided 
by twice the versed sine. Also, for '^ its jperpendictilar to the 
centre of the circle," substitute, the quotient of the square of the 
chord of half the arc divided by twice the versed sine, minus 
the versed sine. 

When the arc is small the curve approximates that of a 
parabola. In this case the equation for the area of the par- 
abola, which is quite simple, may be used. It is this — 

M^-cv. (172). 

3 

Or, in segments of circles where the versed sine is small in 
comparison with the chord, tJie area equals approximately two 
tJiirds of the chord into the versed sine. 



SECTION XII.-THE ELLIPSE. 

451.— Ellipse : Definitions.— Let two lines, PF, PF' {Fig. 
299), be drawn from any point P to any two fixed points 
FF', and let the point P move in such a manner that the sum 
of the two lines, P F, PF', shall remain a constant quantity ; 
then the curve PMKO G A D B P, traced by P, will be an 
EUipse ; the two fixed points F, F' , the Foci ; the point C at 




Fig. 299. 

the middle of FF' , the centre ; the line A J/ drawn through 
F F' and terminated by the curve, the Major or Transverse 
Axis ; the line B O, drawn through C and at right angles to 
A M, the Minor or Conjugate Axis; the line G P, drawn 
through Pand (f and terminated by the curve, the Diameter 
to the point P; the line B K drawn through C, parallel with 
the tangent P T, and terminated by the curve, the diameter 
Conjugate to PG^; the line EUR drawn parallel with DK 
is a double ordinate to the abscissas 6^// and HPoi the di- 
ameter G P [E H — HR) ; the line J L drawn through /^at a 



482 



THE ELLIPSE. 



right angle to A M and terminated by the curve, the Param- 
eter, or Latus Rectum. 

When the point P reaches and coincides with B, the two 
lines P F 2ir\d PF' become equal. 

The proportion between the major and minor axes de- 
pends upon the relative position of F,F\ the foci ; the nearer 
these are placed to the extremities of the major axis the 
smaller will the minor axis be in comparison with the major 
axis. The nearer /% F' approach C, the centre, the nearer 
will the minor axis approach the length of the major axis. 
When F^ F' reach and coincide with the centre, the minor 
axis will equal the major axis, and the ellipse will become a 




circle. Then we have PF — PF' = B C= A C. From this 
we learn PF+ PF' z=^2AC=AM; also, when PF= PF', 
then PF=BF:=:z AC. 

From this we may, with given major and minor axes, 
find the position of i^and F'. To do this, on B, as a centre, 
with A C for radius, mark the major axis at /^and F' . 

452. — Ellipse : Eqiiatloiis to tlie Cwrve. — An equation to 
a curve is an expression containing factors two of which, 
called co-ordinates, measure the distance to any point in the 
curve. For example : in a circle it has been shown {Art. 
443) that P N\s 2i mean proportional to A iV^and N B. Or, 
putting X — A N,y — PN, and a—AB, we have — 

AN : PN : : PN : NB, 



or — 
or — 



J/' =: X {a — x). 



EQUATION TO THE ELLIPSE. 483 

This is the equation to the circle having the origin of x 
and J, the co-ordinates at y4,the vertex of the curve. It will 
be observed that the factors are of such nature in this equa- 
tion, that it may be employed to measure the distance, rect- 
angularly, to P, wherever in the curve the point P may be 
located. By this equation the rectangular distance to any 
and every point in the curve may be measured ; or, having 
the curve and one of the lines xox y, the other may be com- 
puted. 

From this example, the nature and utility of an equation 
to any curve may be understood. The equation to the 
ellipse having the origin of co-ordinates at the vertex, is 
similar to that for the circle. In the form usually given by 
writers on Conic Sections, it is — 



y' = ^,(2ax-x''\ (173.) 



in which a — A C {Fig. 299) \b — B C\ x equals A N, and y = 
PN. 

If, as before suggested, the loci be drawn towards the cen- 
tre and finally made to coincide with it, the minor axis would 
then become equal to the major axis, changing the ellipse into 
a circle. In this case, the factors a and b in the equation would 

become equal; and the fraction —^ would equal — ^ = i,and 

hence the equation would become — • 

y"^ z^ 2 a X — x"^, 
or — y'' ^=z X {2 a— x)\ 

precisely the same as in the equation to the circle above 
shown. The 2 a oi this equation is equivalent to a of the 
circle ; for a in the ellipse represents o*nly half the major 
axis ; while in the equation to the circle a represents the 
diameter. The relation between the ellipse and the circle 
is thus shown ; indeed, the circle has been said to be an 
ellipse in its extreme conditions. 



484 



THE ELLIPSE. 



453.— Ellipse : Relation of Axis to Abscissas of Axes. — 

Multiplying equation (173.) by ^' we have— 

or— a'y' = b' x(2a-x). 

These four factors may be put in a proportion, thus — 

a' \ y :: x{2a-x) : y\ 
representing— 

'AC" ' Tc' '• ANx NM : PN\ 

Or : The rectangle of the two parts into which the ordinate 
divides the axis major is in proportion to the square of the 
ordinate, as the square of the semi-axis major is to the square 
of the semi-axis minor. 




It is shown by writers on Conic Sections that this rela- 
tion is found to subsist, not only with the axes and ordinate, 
but also between an ordinate to any diameter and the ab- 
scissas of that diameter ; for example, referring to Fig. 299 

-CP^-.CD' :: GHy^HP: eTh'. 

If AB' P'M {Fig. 301) be a semi-circle, then {Art. 443) 
pTj;^' = ANxNM. 

Substituting this value oi A NxN M in — 



we have- 



AC - B C -' P'N ■ PN' 
A C : BC :: P'N : FN: 





CF' 


= 


BF' 


— 


BC\ 




x" 


= 


a-" - 


b' 




a' 


-x" 


■= 


a"" - 


{a 


'-y) 






— 


a"" - 


a' 


+ b\ 



RELATION OF TANGENT TO AXIS. 485 

Or : The ordinate in the circle is m. proportion to its correspond- 
vtg ordinate in the ellipse, as the semi-axis major is to the semi- 
axis minor, or as the axis major is to the axis minor. 

454. — Ellipse : Relation of Parameter and Axe§. — The 

equation to the ellipse when the origin of the co-ordinates 
is at the centre is, as shown by writers on Conic Sections, 
thus — 

a'y' = a'b'-b' x'\ (174.) 

or— a'y' = b' {a' - x"). 

If x' equal C F {Fig. 299) then the ordinate will be located 
at^y, and— 



Then- 



This is shown also by the figure. 

Substituting in the above this value oi a"^ — x'^, we have — 

a' y' = b' b' = b\ 

From which, taking the square root— 

ay = b\ 

or — a : b : : b : y. 

Nowjj/, located at Fjf, is the semi-parameter; hence we 
have the semi-minor axis a third proportional to the semi- 
major axis and the semi-parameter. Or : The parameter is a 
third proportional to the two axes of an ellipse. 

455. — Ellipse : Relation of Tangent to the Axes.— Let 

T T' {Fig. 301) be a tangent to P, a point in the ellipse ; then, 
as has been shown by writers on Conic Sections — 

CNxCT=-cm\ 
or— CM : CT :: CN : CM. 



486 THE ELLIPSE. 

Or : The semi-major axis is a mea^i proportional between the ab- 
scissa C N and C Z", the part of the axis intercepted between the 
centre and the tangent. 

This relation is found also to subsist between the similar 
parts of the minor axis ; for— 

CN' xCT' = -cb". 

This relation affords an easy rule for finding the point T, 
or T' ; for from the above we have — 











CT = 


CM' 

' CN 


or, 


putting 


t for 


CT, 


we have 


a" 

x' 


Oi- 








t' = 


b^ 



(175.) 



(176.) 

Since the value of / is not dependent upon y nor upon b, 
therefore / is constant for all ellipses which may be de- 
scribed upon the same major dods A M\ and since the circle 
is an ellipse {Art. 452) with equal major and minor axes, 
therefore rule (175.) is applicable also to a circle, as shown 
in Fig. 301. 

The equation (175.) gives the value of t — C T. From 
this deducting C N = x', we have N T, the subtangent, or — 

CT - CN^ N T, 
t — x' — s\ 
or, substituting for / its value in (175.), we have — 

a 



X 



-^-; (1 77-) 



Or: The subtaiigeyit to an ellipse equals the difference between 
the quotient of the square of the semi-major axis divided by the 
abscissa, and the abscissa ; the origin of the co-ordinates being 
at C, the centre. 



AXES TO CONJUGATE DIAMETER. 



487 



455._Ellip§e : Relation of Tangent with the Foci. — Let 

the two lines from the foci to P {Fig. 302), any point in the 
ellipse, be extended beyond P. With the radius P F' de- 




FiG. 302. 

scribe from P the arc F' G, and bisect it in H. Then the 
line P T, drawn through H, will be a tangent to the ellipse 
at P. 

This has been shown by writers on Conic Sections. The 
construction here shown affords a ready method of drawing 
a tangent. And from the principle here given we learn 
that a tangent makes equal angles with the lines from the 
tangential point to the two foci. 

For, because GH = H F\ we have the angle F' PH = 
H PG. The angles H PG and KPF are opposite, and 
hence {Art, 344) are equal ; and since the two triangles 
F' PHdindKPFdiYQ each equal to HPG, therefore F' PH 
and iTPF are equal to each other. Or: A tangent to an 
ellipse makes equal angles with the two lines drawn from the 
point of tangency to the two foci. 

Experience shows that light shining from one focus is 
reflected from the ellipse into the other focus. It is for this 
reason that the two points Fand F' are called foci, the plu- 
ral of focus ^ a fireplace. 

457- — Ellipse : Relation of Axes to Conjugate Diame- 
ters.— Parallel with K T {Fig. 302) let DEhQ drawn through 



488 THE ELLIPSE. 

C, the centre, and L Q through J, one end of the diameter 
from the point P. Parallel with this diameter PJ draw L K 
and QR through the extremities of the diameter D E. Then 
Z>^ is a diameter conjugate to the diameter PJ, and KR, 
RQ, QL, and Ziif are tangents at the extremities of these 
conjugate diameters. 

Now it is shown by writers on Conic Sections {Fig. 302) 
that— 

ZT' + Wc'' = 'nc' + Yc^ 

or — 

Or : The sum of the squares of the two axes equals the sum 
of the squares of any two conjugate diameters. 

From this it is also shown that the area of the parallelo- 
gram K C equals the rectangle A C x B C; or, that a paral- 
lelogram formed by tangents at the extremities of any two 
conjugate diameters is equal to the rectangle of the axes. 

458- — Ellipse ; Area. — Let E equal the area of an ellipse ; 
A the area of a circle, of which the radius a equals the semi- 
major axis of the ellipse, and let b equal the semi-minor axis. 
Then it has been shown that — 

E : A '.: b '. a, 

E = At 
a 

The area of a circle {Art. 448) is — 

A — \ n dr -^ It r'^, 
and when the radius equals a — 

A ^ 7t a"-, 
This value of A, substituted in the above equation, gives — 

E = 7ta''-, 
a 

E — n ab. ' (178.) 



PRACTICAL SUGGESTIONS. 



489 



Or: The area of an ellipse equals 3- 141 59 J times the product 
of the semi-axes ; or 0-7854 times the product of the axes. 

459. — Ellipse : Practical §ugg[cstioiis. — In order to de- 
scribe the curve of an ellipse, it is essential to have the two 
axes ; or, the major axis and the parameter ; or, the major 
axis and the focal distance. 

If the two axes are given, then with the semi-major axis 
for radius, from B {Fig. 299) as centre an arc may be made 
at i^and F' , the foci ; and then the curve may be described 
by any of the various methods given at Arts. 548 to 552. 

If the major axis only and the parameter are given, then 
{Art. 454) since — 

b^'^ay, 
we have 

b — Vay. (179-) 

Or : The semi-minor axis of an ellipse equals the square root 
of the product of the semi-major axis into the semi-parameter. 
Then, having both of the axes, proceed as before. 

If the major axis and the focal distance are given, or the 
location of the foci ; then with the semi-major axis for ra- 




dius and from the focal points as centres, describe arcs cut- 
ting each other at B and O {Fig. 299). The intersection of 
the arcs gives the limit to B O, the minor axis. With the 
two axes proceed as before. Points in the curve may be 
found by computing the length of the ordinates, and then 
the curve drawn by the side of a flexible rod bent to coin- 
cide with the several points. 

For example, let it be required to find points in the 
curve of an ellipse, the axes of which are 12 and 20 feet ; or 



490 THE ELLIPSE. 

the semi-axes 6 and lo feet, or 6 x 12 = 72 inches, and 10 x 
12 = 120 inches. 

Fix the positions of the points N N', etc., along the semi- 
major axis C 31 {Fig. 303) at any distances apart desirable. 
It is better to so place them that the ordinates when drawn 
shall divide the curve ^Pt^ into parts approximately equal. 
If CM be divided into eight parts as shown, these parts 
measured from C will be well graded if made equal severally 
to the following decimals multiplied by CM. In this case 
CM= 120; therefore — 

^A^ =z 120 X 0-3 =■ 2>^' — x' 
C N' — 120 X 0-475 = 57- = x' 

C N" :=^ 120 X 0'62i, = 75. =y 

Etc., = 120 X 0-75 = 90- = x' 

120 X 0-85 = 102- = x' 

120x0-925 = III- = .f ' 
120 x 0-975 =: 117- — x' 

120 X I -O = 120- = ;r'. 

The equation of the ellipse having the origin of co-ordi- 
nates at the centre {Art. 454) is — 



or, dividing by a 



a'y" = b^{a^-x'\ 



y' = --M'-^'% 



or — 



/ 7,2 

y -^ \ —Aa^ - x"'^ 






°^— y^-^/ a'^-x'^'-, (180.) 

in which a and b represent the semi-axes. Substituting for 
these their values in this case, we have — 

72 



y=-^o ^'2o^- 



j^ = o-6 1/14400 



•/ 2 



LENGTH OF ORDINATES. 



491 



Now, substituting in this equation the several values of 
x'^ successively, the values of the corresponding ordinates 
will be obtained. For example, taking 36, the first value of 
x\ as above, we have — 

y= 0-6 1/14400 



36^ 



7 = 68-684; 



j; — 0-6 1^14400 - 57^ 
y = 63-359; 

and so in like manner compute the others. 

The ordinates for this case are as follows, viz. : 



When .v 

X 



— o, _y = 72-0 

r:= 36,7 = 68.684 

= 57.7=^63.359 

=. 7Sy y = 56-205 

= 90, JJ/ == 47-624 

= 102, J/ = 37.928 

= III, y =z 27.358 
= 117, J/ = 15-999 
= 120, J 1= o-o. 



The computation of these ordinates is accomplished easi- 
ly by the help of a table of square roots and of logarithms. 

For example, the work for one ordinate is all comprisjd 
within the following, viz. : 



7 



0-6 r 14400 
120- = 14400 
36- = 1296 



36- = 68.684. 



13104 = 4. 1 174039 

Half = 2-0587020 

0.6 = 9.7781513 

68.684 = 1-8368533. 

The logarithm of 13104 = 4-1 174039. The half of this is 
the logarithm of the square root of 13 104. To the half log- 
arithm add the logarithm of o-6; the sum is the logarithm 
of 68-684 found in the table (see Art. 427). 



SECTION XIII.— THE PARABOLA. 

460. — Parabola : I>efinitioiis. — The parabola is one of 
the most interesting of the curves derived from the sections 
of a cone. The several curves thus produced are as fol- 
lows : When cut parallel with its base the outline is a circle ; 
when the plane passes obliquely through the cone, it is an 
ellipse ; when the plane is parallel with the axis, but not in 
the axis, it is a hyperbola ; while that which is produced by 




304. 



a plane cutting it parallel with one side of the cone is a 
parabola. 

Let the lines L M and L N (Fig. 304) be at right angles ; 
draw CFB parallel with LM; make LQ = LF\ draw QB 
parallel with LF\ then FB — B Q. Now let the line A L 
move from F L^ but remain parallel with it, and as it moves 
let it gradually increase in length in such manner that the 
point A shall constantly be equally distant from the line LM 
and from the point F. Then A BP, the curve described by 
the point A, will be a semi-parabola. For example, the 
lines FB and B Q are equal ; the lines FPand PM^vq equal, 
and so of lines similarly drawn from any point in the curve 
A BP. Let PNhQ drawn parallel with LM; then for the 



EQUATION TO THE CURVE. 493 

point P, AN IS the abscissa and NP its ordinate {see Ar/. 
452). 

The double ordinate CB drawn through F, the focus, is 
the para^neter. A F is the focal distance. A is the vertex of 
the curve. The line L M is the directrix. 

4-61. — Parabola : Equation to tlie Curve. — In Fig, 304 
F PN is a right-angled triangle, therefore — 



NP"" = FP"" - FN''; 
but- FP^ MP= LN=A N-\-AL', 

and— FN= AN- AF. 

Therefore — 



NP' = A.N-hA L'-A N- A F' ; 

or- _^.^=.(.^^ + i/)^-Gr-l/)^ 

/ being put for the distance LF— FB (see Art. 452). As 
in Arts. 412 and 413, we have — 

{x-v \py = ,r"' ^px-Y\p'' 

{x-^py^x^-px+ip-' 

y'' — 2px (181.) 

by subtraction. This is the usual equation to the parabola, 
in which we have the rule : The square of the ordinate equals 
the rectangle of the corresponding abscissa with the parani- 
eter. 

From (181.) we have — 

X : y : : y '. 2p, 

or: Th.e parameter is a third proportional \,o the abscissa and its 
corresponding ordinate. 

462. — Parabola ; Tangent. — From AT, any point in the 
directrix, draw a line to F, the focus {Fig. 305) ; bisect M F 
in R, and through R draw U T perpendicular to M F, then 
the line T U will be a tangent to the curve. For, draw AI D 



494 



THE PARABOLA. 



perpendicular to L V, and from P, the point of its intersection 
with the line T [/, draw a line to F, the focus ; then, because 
RP is a. perpendicular from the middle of MF, MPF is an 
isosceles triangle, and therefore the lines J/Pand FP are 
equal, or the point P is equidistant from the focus and from 
the directrix, and therefore is a point in the curve. 

To show that the line T U touches the curve but does not 
pass through it, take U, any point in the line T U, other than 




Fig. 305. 



the point P\ join U to i/and to F. Then, since f/ is a point 
in the line T U, M U F, for reasons above given, is an isosce- 
les triangle ; from C/draw U F perpendicular to L V. Now, 
if the point ^be also in the curve, the lines C/Fand U F, 
by the law of the curve, must be equal ; but UF, as before 
shown, is equal to UM, a line evidently longer than U V; 
therefore, it is evident that the point ^is not in the curve. 
A similar absurd result will be reached if any other point 
than the point [/ in the line U The assigned, excepting the 



RULE FOR THE TANGENT. 495 

point P. Therefore the line T P touches the curve in only 
one point, P ; hence it is a tangent. 

Parallel with L F, from A, draw ^4 5, the vertical tangent. 
Now ^ vS bisects M F or intersects it in the point i'?. For 
the two right-angled triangles F L J^/and FA R are homolo- 
gous ; and because FA ■=^ A L, by construction, therefore 
FR^RM. 

Or : The vertical tangent bisects all lines which can be drawn 
from the focus to the directrix. 

The lines P/^and FT are equal ; for the lines J/Pand 
N T being parallel, therefore the alternate angles MP T 
and N TP Sire equal (Art. 345) ; and because the line P T bi- 
sects MF, the base of an isosceles triangle, therefore the 
angles ^P 7" and T^P 7^ are equal. We thus have the two 
angles N TP and FPT each equal to the angle MPT; 
therefore the two angles N TP and FP T are equal to each 
other ; hence the triangle PF T is an isosceles triangle, hav- 
ing the points T and P equidistant from F, the focus. 

Also because the line MF is perpendicular to P T, there- 
fore the line MF bisects the tangent PT in the point R. 
And because TR = R P, therefore, comparing triangles 
TRF and TPO, TF= FO. 

The opposite angles MP T and U PD made by the two in- 
tersecting lines U T and M D {Art. 344) are equal, and since 
the angles MP T and FP T are equal, as before shown, 
therefore the angles FP T and U P D are equal. 

It is because these two angles are equal, that, in reflectors, 
rays of light and heat proceeding from F, the focus, are re- 
flected from the parabolic surface in lines parallel with the 
axis. 

For an equation expressing the value of the tangent, we 
have — 



TP = TN + NP\ 

f^ = (2xY+y\ 

Or: The tangent to a parabola equals the square root of tJie 
sum of four times the square of the abscissa added to the square 
of the ordinate. 



496 THE PARABOLA. 

463. — Parabola: Subtangent. — The line T N {Fig. 305), 
the portion of the axis intercepted between T, the point of 
intersection of the tangent, and N, the foot of a perpendicu- 
lar to the axis from P, the point of contact, is the subtangent. 
The subtangent is bisected by the vertex, or TA = A N. 
For, the two triangles TRA and T P N 2irQ homologous; 
and, as shown in the last article, the line jl/i^ bisects P Tin 
R', or TR = RP, 

Therefore, we have — 

TR : TA :: TP : TN, 
TR X TN= TA X T P, 



but— 


TR=i TP; 


therefore — 


^TPx TN=z TA X TP, 




i TN= TA. 



Or : The subtangent of a parabola is bisected by the vertex ; or 
is equal to twice the abscissa. 

And because of the similarity of the two triangles TRA 
and TP N, as above shown, we have — 

NP=^ 2 AR, 

y = 2 AR. 

Or : The ordinate equals twice the vertical tangent. 

464. — ParaboBa: Normal and Subnormal. — The line 
PO {Fig. 305) perpendicular to P T, is the normal and NO, 
the part of the axis intercepted between the normal and the 
ordinate, is the subnormal. For the normal, from similar 
triangles, we have — 

TN : NP :: T P : P O, 

i\^Px TP 
^^ ~ TN ' 

P0 = ^-^. 

2X 



. DIAMETER AND RECTANGLE OF BASE. 497 

Or : The 7iormal equals the rectangle of the ordinate and taji- 
gcnty divided by tivicc the abscissa. 

The subnormal equals half the parameter. For (181.) — 



or— NP"- = 2 FB ' AN. 

Dividing by 2 XN gives— 



NP'- 



2 AN 

In the similar, triangles {Art. 443) 6^/^iVand P T N, we 
have — 

NO : iV/' : : NP : NT, 

NO^EIl. 
NT 

As shown in the previous article, N T — 2 A N\ therefore — 

NO^Z^. (B.) 

2AN 

Comparing equations (A.) and (B.), we have — 

NO=^FB. 

Or : The subnormal of a parabola equals Jialf the parameter, 
a constant quantity for the subnormal to all points of the 
curve. 

465 — Parabola: Diameters. — In the parabola BAC* 
{Fig. 306), PD, a diameter (a line parallel with the axis) to 
the point P, is in proportion to B T> x B C, the rectangle of 
the two parts into which the base of the parabola is divkled 
by the diameter. 

This may be shown in the following manner:. 

nP=£N:=FA-NA. (A.) 



498 THE PARABOLA. 

For^^ we have, taking the co-ordinates, for the point 
C, (i8i.)— 

or — -^ = X 

or- ^- ^EA. (B.) 




For N A we have, taking the co-ordinates to the point 

or — -—- — X, 

2p 



N P'' 

or— -—---^NA, (C.) 

Using these values (B.) and (C.) in (A.), we have — 
D P=EA —NA, 



EC NP' _ EC'-NP' 

~ 2p 2/ ~ 2/ 

If / be put for B C and n for D C, then — 
NP=EC-DC=^l- 72. 



TO FIND THE CONSTITUENT PARTS. 499 



and — 


2/ 


then {Art. 412,)- 


— 




2/ 


or {Art. 415) — 


2/ 




2/ 




2/ ' 




DP- DCxBD 



2/ 

Now, since 2 /, the parameter, is constant, we have D P, 
the diameter, in proportion to D C x B Dy the two parts of 
the base. 

Putting d for the diameter, we have — 

d='l^-J^, (183.) 

2/ 

Or : The diameter of a parabola equals the quotient obtained 
by dividing th.Q rectangle — formed by the two parts into which 
the diameter divides the base — by the parameter. 

It has been shown by writers on Conic Sections that a 
diameter, P J {Fig, 307), to any point Pm a parabola bisects 
all chord lines, SG,B£, etc., drawn parallel with the tan- 
gent to the point P; the diameter being parallel with the axis 
of the parabola. 

466. — Parabola : Elemems. — From any given parabola, 
to find the axis, tangent, directrix, parameter and focus, 
draw any two parallel lines or chords, SG and DE (Fig. 
307), and bisect them in N and J; through these points 
draw JP; then y/'will be a diameter of the parabola — a 



500 



THE PARABOLA. 



line parallel with the axis. Perpendicular to Py draw the 
double ordinate P Q and bisect it in A^; through iVand par- 
allel with Py draw TO, cutting the curve in A ; then TO 
will be the axis. Make AT^AN, join Tand P\ then TP 
will be the tangent to the point P; from P draw P O per- 
pendicular to P T \ then P O will be the normal, and NO the 
subnormal. 

With NO for radius, from N as a centre, describe the 
quadrant O Ps.\ draw R C parallel with A O, cutting the curve 




Fig. 307. 

in C; from C draw CB perpendicular to A O, cutting A O in 
F\ then Pwill be the focus and C B Xh^ parameter. Make 
AL — AF\ draw L M perpendicular to T0\ then Z J/ will 
be the directrix. Extend PJ to meet Z J/at M', join Pand 
F\ then, if the work has been properly performed, F P will 
equal MP. 

467. — Parabola: Described Meeliaiiically. — With NP 
{Fig. 308) a given base, and N A a given height, set perpen- 



THE CURVE DESCRIBED MECHANICALLY. 



;oi 



dicularly to the base, extend N A beyond A, and make A T 
equal to N A ; join Zand P; from /^perpendicularly to TP 
drawP6^; bisect ON\\\R\ make AL and y^/'each equal 
X.O N R\ through Z, perpendicular to L O, draw D E, the di- 
rectrix. 

Let the ruler CDBShe laid to the line DE, then with 
J G H, ^ set-square, the curve may be described in the fol- 
lowing manner : 

Placing the square against the ruler and with its edge 




Fig. 30S. 



y// coincident with the line MP, fasten to it a fine cord on 
the edge PR, and extend it from P to F, the focus, and se- 
cure it to a pin fixed in F. The cord FP will equal the 
edge MP. To describe the curve set the triangle J G H at 
MPE, slide it gently along the ruler towards D, keeping the 
edge y G in contact with the ruler, and, as the square is 
moved, keep the cord stretched tight, holding for this pur- 
pose a pencil, as at K, against the cord. Thus held, as the 
square is moved the pencil will describe the curve. That 
this operation will produce the true curve we have but to 



502 



THE PARABOLA. 



consider that at all points the line F K will equal KJ, which 
is the law of the curve (Art. 460). 

468. — Parabola : Described from Points. — With given 
base, N P (Fig. 309), and given height, A N, to find the points 
D, F, M, etc., and describe the curve. Make A T equal to 
A N (Art. 462) ; join T and P\ perpendicular to TP draw 




Fig. 309. 



PO; make A B equal to twice N 0\ take G, any point in the 
axis A O, and bisect B G in J \ on y as a centre describe the 
semi-circle B CG cutting A Z, a perpendicular to B O in C ; 
on A C and A G complete the, rectangle A C D G. Then D is 
a point in the curve. Take H, another point in the axis ; 
bisect B H\n K] on K as a centre describe the semi-circle 
B E H cutting A L\xi E\ this by E F and HE, gives F, an- 



THE CURVE DESCRIBED FROM ARCS. 503 

Other point in the curve ; in like manner procure M, and as 
many other points as may be desired. This simple and 
accurate method of obtaining points in the curve depends 
upon two well-established equations ; one, the equation to 
the parabola, and the other, the equation to the circle. The 
line G D, an ordinate in the parabola, is equal to A C, an 
ordinate in the circle BCG\ AG, the abscissa of the para- 
bola, is also the abscissa of the circle ; in which we have 
{Art. 443)- 

AG \ AC \\ AC : AB, 
X : y : : y : a — Xy 
j/^ =z x(a — x). 
For the parabola, we have (181.) — 

JK" = 2pX. 

Comparing- these two equations, we have— 

X {a — x) = 2px, 

a — ;ir = 2 /, 
or — 

BG~AG=2p. 

By construction AB equals 2 N O, or twice the subnor- 
mal ; the subnormal {Ai't. 464) equals half the parameter. 
Hence, twice the subnormal equals the parameter — equals 
2 p. Therefore, the method shown in Fig. 309 is correct. 

4-69- — Parabola: Described from Arc§. — Let NP {Fig. 
310) be the given base and A iVthe given height of the par- 
abola. Make A T {Art. 462) equal A N. Join TtoP\ draw 
P O perpendicular to P T\ bisect N O \n R\ make A L and 
A F each equal to N R ; then L M, drawn perpendicular to 
TO, will be the directrix. Parallel to LM draw the lines 
B D, C E, etc., at discretion. Then with the distance B L for 



504 



THE PARABOLA. 



radius, and on i^ as a centre, mark the line B D with an arc ; 
the intersection of the arc and the line will be a point in the 
curve {Art. 460). Again, with C L for radius and on i^as a 
centre, mark the line C E with an arc ; this gives another 
point in the curve. In like manner, mark each horizontal 




Fig, 3ro, 

line from /^ as a centre by a radius equal to the perpendicu- 
lar distance between that line and L M, the directrix. Then 
a curve traced through the points of intersection thus ob- 
tained will be the required parabola. 



470. — Parabola: De§cribed from Orc1iiiate§. — With a 
given base, N P {Fig. 311), and height, A N, ci parabola may 
be drawn through points J, H, G, etc., which are the extrem- 
ities of the ordinates B J, C H, D G, etc. ;. the lengths of the 
ordinates being computed from the equation to the curve, 
(181.)- 

jk' = 2px. 

For any given parabola, in base and height, the value of 



THE CURVE FORMED BY ORDINATES. 



505 



/ may be had by dividing both members of the equation by 
2 ;ir; by which we have — 



p = ^~.- 



NP' 



2X 2AN' 



(A.) 



from which, TVPand A iV being known,/ may be computed. 
With the value of/, a constant quantity, determined, the 
equation is rendered practicable. For, taking the square 
root of each member of equation (181.), we have — 

y= V 2p X. (B.) 

which by computation will produce the value of jf, for every 
assigned value of ;r, 2iS> A B, A C, A D, etc. 




Fig. 311. 



As an example : let it be required to compute the ordi- 
nates in a parabola in which the base, N P, equals 8 feet, and 
the height, A N, equals 10 feet. With these values equation 
(A.) as above becomes — 



_ NP^ 

^ ~ Jan 



2 X 10 



20 



2 p — 6-4. 



5o6 



THE PARABOLA. 



Then, with this value in (B.) as above, we have, for each or- 
dinate — 



J= V6 



4x. 



In order to assign values to x, let A N be divided into 
any number of parts at B, C, D, etc., say, for convenience in 
this example, in ten equal parts ; then each part will equal 
one foot, and we shall have the consecutive values oi x = i, 
2, 3, 4, etc., to ID, and the corresponding values of 7 will be 
as follows. When — 



I, J^= V6 



X — 2, J/ = 4/6 

X 



3, J = \'6 



X :=^ 4, f = V6 

X 



5, J= V6 
X =z 6, y = ^'-^ 



X ^ 8, J = |/6 



X = 10, y = |/6 



4x 



I = V 6.4= 2.5297 = ^7, 



3-5777 = CH, 



4 X 2 = |/ 12- 

4x 3= v' 19-2 = 4-3^18 = Z> 6^, 

4x 4= |/ 25-6 = 5-0596 = ^^, 
= 5-6569 = (etc.). 



4 X 5=1/32 



4x6=1/ 38-4 = 6.1968 = 
■ 1/^4^8 = 6-6933 = 



4x 7 



4x8=4/ 51-2 = 7-1554 
4x9= |/l7^ = 7-5895 



4x10— '^Z 64 



8-0 



^NP. 



With these values of y, respectivelv, set on the correspond- 
ing horizontal lines B J, CH, DG, £ S, etc., points in the 
curve y, //, G, 5, etc., are obtained, through which the curve 
may be drawn. The decimals above shown are the decimals 
of a foot ; they may be changed to inches and decimals of an 
inch by multiplying each by 12. For example : 12x0-5297 
= 6-3564 equals 6 inches and the decimal 0-3564 of an inch, 
which equals nearly § of an inch. 

Near the top of the curve, owing to its rapid change in 
direction and to the approximation of the direction of the 
curve to a parallel with the direction of the ordinates, it is 



THE CURVE FOUND BY DIAMETERS. 



507 



desirable to obtain points in the curve more frequent than 
those obtained by dividing the axis into equal parts. 

Instead, therefore, of dividing the axis into equal parts, 
it is better to divide it into parts made gradually smaller 
toward the apex of the curve— or, to obtain points for this 
part of the curve as shown in the following article. 

471. — Parabola: Dci^cribed from Diameters. — Let EC 

{Fig. 312) be the given base and A E the given height, placed 
perpendicularly to E C. Divide E C \w several parts at 
pleasure, and from the points of division erect perpendicu- 
lars to E C. The problem is to compute the length of these 
diameters, as DP, and thereby obtain points in the curve, as 
at P. For this purpose we have equation (183.), which gives 




D C 

Fig. 312. 

the length of the diameters, and in which n equals D C {Fig. 
312), /equals twice E C\ and / equals half the parameter of 
the curve. The value of p is given in equation (A.), {Art, 
470), in which y equals EC {Fig. 312), and x equals A E. 
Substituting these symbols in equation (A.), we have — 



/ 



y 



EC 



2x 2xAE 



11 

2 /r 



where h — EC, the base, and h— A E, the height. For /, 
substituting this, its value, in equation (183.), we have — 

71 {I — 11) n {I — ri) 



2p 
h n (2 /; — fi) 



2 h 



(184.) 



508 THE PARABOLA. 

As an example : let it be required in a parabola in which 
the base equals 12 feet and the height 8 feet, to compute the 
length of several diameters, and through their extremities 
describe the curve. Then Ji will equal 8, and b 12. 

If the base be divided into 6 equal parts, as in Fig. 312, 
each part will equal 2 feet. Then we have — 



and- 



IL - 8 _ 8 _ I 
b~^~ ~\2} ~ i44~ 18 



d= -j-^ n {2 b- n)y 



n(24-n) 
a= — — 7: . 



In this equation, substituting the consecutive values of ji, 
we have, when — 

- O X 24 

;/ = o, a = = o 

I o 

2 X 22 



71 = 2, 


a = 


18 


=2.444 


71= 4, 


d = 


4x 20 
18 


=4.444 


71 =r. 6, 


d = 


6x 18 
18 


= 6. 


71 =z 8, 


d = 


8x 16 

1 Q 


=37.111 



lox 14 
12 X 12 

;^=rl2, ^=— Yg ^=:8.0 

The several diameters, as P D, in Fig. 312, may now be 
made equal respectively to these computed values of d, and 
the curve traced through their extremities. 



AREA EQUALS TWO THIRDS OF RECTANGLE. 5Q9 

472. — Parabola: Area, — From (181.), the equation to 
the parabola, and by the aid of the calculus, it has been 
shown that the area of a parabola is equal to two thirds of 
the circumscribing rectangle. For example : if the height, 
AE {Fig. 312), equals 8 feet, and EC, the base, equals 12 
feet, then the area of the part included within the figure 
A PC E A equals f of 8 x 1 2 = | x 96 = 64 feet ; or, it is equal 
to f of the rectangle A B C E. 



SECTION XIV.— TRIGONOMETRY. 

473. — Riglit-Ang^led Triaiigle§ : The Sides. — In right- 
angled triangles, when two sides are given, the third side 
may be found by the relation of equality which exists of 
the squares of the sides (Arts. 353 and 416). For example, 




Fig. 313. 

if the sides a and b {Fig. 313) are given, c, the third side, 
may be computed from equation (115.) — 

Extracting the square root, we have — 

c = Vd-U-a\ 

When the hypothenuse and one side are given, by transposi- 
tion of the factors in (115.), we have — 



a 



c'- b'; 



or- 



a= Vc"- — b 
b' = c 



V'c 



a 



a 



(A.) 



(B.) 



TWO SIDES GIVEN TO FIND THE THIRD. 5 I I 

Owing to the factors being- involved to the second power in 
this expression, the labor of computation is greater than 
that in a more simple method, which will now be shown. 

In equation (A.) or (B.) the factors under the radical may 
be simplified. By equation (i 14.) we have — 

Therefore, equation (A.) becomes — 



rt = y (^ + b) {c — b), 

a form easy of solution. 

For example: let c equal 29-732 and b equal 13-216, then 
we have — 

29-732 

13-216 

The sum = 42-948 
The difference = 16-516 

By the use of a table of logarithms {Art. 427) the problem 
may be easily solved ; thus — 

Log. 42-948 = 1-6329429 
16-516 = I -2179049 

To get the square root — 2)2 - 8508478 

a = 26-6332 = 1-4254239 

This method is applicable to the sides of a triangle, only ; 
for the hypothenuse it will not serve. The length of the 
hypothenuse as well as that of either side may, however, be 
obtained by proportion ; provided a triangle of known di- 
mensions and with like angles be also given. 

For example: in Fig. 314, in which the two sides a and 
b are known, let it be required to find c, the hypothenuse. 

Draw the line D E parallel with A C, then the two trian- 
gles B DE and B A C are homologous; consequently their 



512 



TRIGOXOiMETRY. 



corresponding sides are in proportion (Art. 361). Hence, if 
d equals unity, we have — 

d : / : : a : c, 

= ^/, 

from which, when a and / are known, c is obtained by sim- 
ple multiplication. 

4-74. — Riglit- Angeled Triangles: Trigonometrical Ta- 
bles. — To render the simple method last named available, 
the lengths of <r/, e and / i^^]^- 314) have been computed for 
triangles of all possible angles, and the results arranged in 




FiCx. 314. 

tables, termed Trigonometrical Tables. The lines d, e, and 
/, are known as siucs, cosiuts, tangents, cotangents, etc., as 
shown in Fig. 315 — where A B is the radius of the circle 
B C H. Draw a line A F, from A, through any point, C, of the 
arc B G. From C draw CD perpendicular to A B ; from B 
draw BF perpendicular to A B ; and from G draw 6^ /^per- 
pendicular to A G. 

Then, ior the angle FAB, when the radius y^ (7 equals 
unitv, CD is the sine; AD the cosine ; D B the 7'ersed sine ; 
^F the tangent ; GF the cotangent ; AF the secant ; and 
\ /^ the cosecant. 

But if the angle be larger than one right angle, yet less 
an two right angles, as BAH, extend HA to A" and F B 
K, and from //draw H J perpendicular to A J. 



TRIGONOMETRICAL TABLES. 



513 



Then, for the angle BAH, when the radius A H equals 
unity, H y IS the smc ; A J the cosine ; BJ the versed sine ; 
B K \\\Q tangent ; and A K the secant. 

When the number of degrees contained in a given angle 
is known, the value of the sine, cosine, etc., corresponding to 
that angle, may be found in a table of Natural Sines, Co- 




flG. 315. 

sines, etc. Or, the logarithms of the sines, cosines, etc., may 
be found in logarithmic tables. 

In the absence of such a table, and when the degrees 
contained in the given angle are unknown, the values of 
the sine, cosine, etc., may be found by computation, as fol- 
lows:— Let ABC {Fig. 316) be the given angle. At any 
distance from B draw b perpendicular to B C. By any scale 
of equal parts obtain the length of each of the three lines a, 
b, c. Then for the angle at B we have, by proportion — 



5H 



TRIGONOMETRY. 



c 


: b : 


: i-o : 


sin. 


B 


_b 
c 


c 


: a : 


: I -O : 


COS. 


B 


a 

c ' 


a 


: b : 


: I -O : 


tan. 


B 


_ b 

a ' 


b 


: a : 


: i-o : 


cot. 


B 


a 
~b' 


a 


: c : 


: I -o : 


sec. 


B 


_ c 
a' 


b 


: c : 


: I -0 : 


cosec 


.B 


c 



Or, in any right-angled triangle, for the angle contained 
between the base and hypothenuse — 



When perp. divided by hyp., the quotient equals the sine. 



'* base 


" hyp.. 


" perp. " 


" base. 


base 


" perp. 


" hyp. 


" base, 


" hyp. 


" perp- 



cosine. 

tangent. 

cotangent. 

secant. 

cosecatit. 



To designate the angle to which a trigonometrical term 
applies, the letter at the intended angle is annexed to the 




name of the trigonometrical term ; thus, in the above exam- 
ple, for the sine of ABC we write sin. B : for the cosine, 
cos. B, etc. 



EQUATIONS TO RIGHT-ANGLED TRIANGLES. 



515 



By these proportions the two acute angles of a right- 
angled triangle may be computed, provided two of the 
sides are known. For when the perpendicular and hypoth- 
enuse are known, the sine and cosecant may be obtained. 
When the base and hypothenuse are known, the cosine and 
secant may be computed. And when the base and perpen- 
dicular are known, the tangent and cotangent may be com- 
puted. 

Either one of these, thus obtained, shows by the trigo- 
nometrical tables the number of degrees in the angle ; and, 
deducting the angle thus found from 90°, the remainder will 
be the angle of the other acute angle of the triangle. For 




example : in a right-angled triangle, of which the base is 8 
feet and the perpendicular 6 feet, how many degrees are 
contained in each of the acute angles ? 

Having, in this case, the base and perpendicular known » 
by referring to the above proportions we find that with 
these two sides we may obtain the tangent ; therefore — 



Tan.i? = ^-=|=:0.75. 



Referring to the trigonometrical tables, we find that 0-75 is 
the tangent of 36° 52' 12'^, nearly; therefore — 



The quadrant equals 90- o- o 
The angle B equals 36- 52 -12 

The angle A equals .53 •07-48 



5l6 TRIGONOMETRY. 

476. — Riglit-Aiigled Triangles: Trigonometrical Value 
of Sides.— In the triangle A B C {Fig. 317), with D P^\ for 
radius, and on ^ as a centre, describe the arc P D, and from 
its intersection with the lines A B and B C, draw PM and 
TD perpendicular to the line B C. Then from homologous 
triangles we have these proportions for the perpendicular — 

BD : D T :: BC : C A, 

r : tan. B : : base : perp., 

I L tan. B w a \ b — a tan. B. (185.) 

Also— 

BP : PM :: BA : AC, 

r : sin. B : : hyp. : perp., 

I : sin. B : : c : d = c sin. B. (186.) 

For the base, we have — 

BP : BM :: BA : BC, 

r : cos. B : : hyp. : base, 

I : COS. B \ \ c \ a — c cos. B. (187.) 

Again- — 

TD : BD '.'. AC \ BC, 

tan. B \ r \ \ perp. : base, 

tan. B \ I wb'.a — -, (188.) 

tan. B ^ ^ 

For the hypothenuse, we have — 

PAI '. PB :: AC :: AB, 
sin. B \ r \\ perp. : hyp., 



RULE FOR THE PERPENDICULAR. 517 

sin. B : i w b -. c — -. --. (i8q.) 

sin. B 

Again — 

BD : B T :: BC : B A, 

r : sec. B : : base : hyp., 



I : sec. B : : a : c = a sec. B — . (iQO.) 

COS. B ^ ^ ^ 



This substitution of the cos. for the sec. is needed because 
tables of secants are not always accessible. That it is an 
equivalent is clear ; for we have — 



BM : BP :: BD : B T, 



I 
cos. \ r \ \ r \ sec. =: 

COS. 



By these equations either side of a right-angled triangle 
may be computed, provided there are certain parts of the 
triangle given. As, for example : of the six parts of a tri- 
angle (the three sides and the three angles), three must be 
given, and at least one of these must be a side. 

As an example : let it be required to find two sides of a 
right-angled triangle of which the base is 100 feet, and the 
acute angle at the base is 35 degrees. Here we have given 
one side and two angles (the base, acute angle, and the right 
angle) to find the other two sides, the perpendicular and the 
hypothenuse. 

Among the above rules we have, in equation (185.), for 
the perpendicular — 

h ^=:^ a tan. B. 

Or: TMq perpendicular equals t\\c product of the base into the 
tangent of the acute angle at the base. 



5lo TRIGONOMETRY. 

Then (^;Y. 427) — 
The logarithmic tangent of B {— 35°) is 9-8452268 
Log. of <^ (= 100) is 2 -0000000 

Perpendicular, b (= 70-02075) = i -8452268 

And for the hypothenuse, taking equation (190.), we 
have — 






COS. B 



Or : The Jiypothcmise equals the quotient of the base divided 
by the cosine of the acute angle at the base. 
For this we have — 

Log. of a (= 100) is 2-0000000 

'' COS. B (= 35°) is 9-9133^45 

H3^pothenuse c{— 122-0775) = 2-0866355 

We thus find that a right-angled triangle, having an angle 
of 35 degrees at the base, has its three sides, the perpendic- 




ular, base, and hypothenuse, respectively equal to 70-02075, 
100, and 122-0775. 

N.B. — The angle at A {Fig.'^if) is obtained by deducting 
the angle at B from 90° {Art. 346). Thus, 90 — 35 = 55 ; 
this is the angle at y^, in the above case. 

If the perpendicular be given, then for the base use 
equation (188.), and for the hypothenuse use equation (189.). 
If the hypothenuse be given, then for the base use equation 
(187.') and for the perpendicular use equation (186.). 



USEFUL RULE FOR THE SIDES. 519 

4-76. — ObBique-Aiigflecl Triangles: Sines and Sides. — In 

the oblique-angled triangle A B C {Fig. 318) from C and per- 
pendicular to A B draw CB. This line divides the oblique- 
angled triangle into two right-angled triangles, the lines and 
angles of which may be treated by the rules already given ; 
but there is a still more simple method, as will now be 
shown. 

As shown in Art. ^j\\ ^'When the perpendicular is di- 
vided by the hypothenuse the quotient equals the sine." 
Applying this to Fig. 318, we have — 

sm. A = -', 
b 

sm. j5 = — . 



Let the former be divided by the latter ; then — 

d 
sin. A b 





sin. B d 




a 


or, reducing, we have- 


- 




sin. A _a 



sin. B b 

or, putting the equation in the form of a proportion — 

sin. B : sin. A : : b : a; 

or ; the si?ics are in proportion as the side's, respectively 0/?- 
positc. Or, as commonly stated, the sines are in proportion 
as the sides which subtend them. 

This is a rule of great utility ; by it we obtain the follow- 
ing : 

Referring to Fig. 318, we have — 

. „ . . , , sin. A f . 

sm. B : sm. A : : b : a = b . (iQi-) 

sm. B 



520 TRIGONOMETRY. 

^ . sin. A , . 

sin. 6 : sin. ^ : : ^ : ^ = ^-^ . (IQ2.) 

sin. C 

/I • r> 7 sin, B , . 

sin. A : sin. B : : a : b =. a -r- — -- . ( IQ^.) 

sin. A ^ ^^ ^ 

^ • -n 7 sin. B , . 

sin. 6 : sin. B \ \ c \ b ^^ c —. — . {IQ4.) 

sin. C ^ ^-^ J 

. . ^ sin. C , . 

sin. A : sin. 6 : : a : c = a —. . iiQ^.) 

sm. A ^ ^^ ^ 

sin. B : sin. C : : b : c = b . ' . (iq6.) 

sin. j5 ^ ^ / 



These expressions give the values of the three sides respec- 
tively ; two expressions for each, one for each of the two 
remaining sides ; that is to be used which contains the ^zven 
side. 

From these expressions we derive the values of the 
sines; thus — 

(I97-) 
(198.) 

(199-) 

(200.) 



sin. 


A = sin. 


^-■ 


sin. 


A = sin. 


c 


sin. 


B = sin. 


a'-. 

a 


sin. 


B = sin. 


ct. 

c 



sin. C = sin. A -. (201.) 



sin, C = sin. B -. (202.) 



477. — Oblique - Angled Triangles: First Class. — The 

problems arising in the treatment of oblique-angled trian- 
gles have been divided into four classes, one of which, the 



TO FIND THE TWO SIDES. 52 1 

first, will here be referred to. The problems of the first 
class are those in which a side and tzvo angles are given, to 
find the remaining angle and sides. 

As to the required angle, since the three angles of every 
triangle amount to just two right angles {Art. 345), or 180°, 
the third angle may be found simply by deducting the sum 
of the two given angles from 180°. 

For example : referring to Fig. 318, if angle ^ = 18° and 
angle B — 42°, then their sum is 18 + 42 = 60, and 180 — 
60 = 120° = the angle A C B. 

To find the tw^o sides : if ^ be the given side, then to find 
the side b we have, equation (193.) — 



sm. B 

^ A 

sm. A 



or, the side b equals the product of the side a into the quo- 
tient obtained by a division of the sine of the angle opposite 
b by the sine of the angle opposite a. 

For example: in a triangle {Fig. 318) in Avhich the angle 
A = 18°, the angle B — 42° (and, consequently {Art. 345) the 
angle C= 120°), and the given side a equals 43 feet; what 
are the lengths of the sides b and c? Equation (193.) gives — 

, sin. B 

b = a— — —J. 
sm. A 

Performing the problem by logarithms {Art. 427), we 
have — 

Log. a (= 43) =^-6334685 
Sin. B{— 42°) = 9-8255109 

£•4589794 
Sin. A (= 18°) = 9.4899824 

Log. ^ (= 93 • 1102) = 1 .9689970. 

Thus the side b equals 93.1102 feet, or 93 feet i inch and 
nearly one third of an inch. 



522 TRIGONOMETRY. 

For the side c, we have, equation (195.)- 

sin. C 



c ^ a 



sin. A ' 



or- 



Log. «(=43) =£-6334685 
Sin. C{= 120°) = 9-9375306 

£•5709991 
Sin. A (= 18°) ^ 9.4899824 

Log. ^(== 120-508) =: 2-0810167 

or, the base c equals 120 feet 6 inches and one tenth of an 
inch, nearly. But if instead of a the side b be given, then 
for a use equation (191.), and for c use equation (196.). 

And, lastly, if c be the given side, then for a use equation 
(192.), and for b use equation (194.)- 



478. — Oblique-Angled Triangles: Second Class. — The 

problems which comprise the second class are those in which 
tivo sides and an angle opposite to one of them are given, to 
find the two remaining angles and the third side. 

The only requirement really needed here is to find a 
second angle ; for, with this second angle found, the problem 
is reduced to one of the first class ; and the third side may 
then be found under rules given in Art. 477. 

To find a second angle, use one of the equations (197.) to 
(202.). 

For example : in the triangle ABC {Fig. 318), let a(j=. 43) 
and ^(=93-11) be the two given sides, and A, the angle op- 
posite a, be the given angle (= 18°). Then to find the angle 
B, we have equation (199.) — (selecting that which in the 
right hand member contains the given angle and sides) — 

sin. B = sin. A - 
a 

= sin.^?llLi. 
43 



OBLIQUE-ANGLED TRLVNGLES. 523 

By logarithms {Art. 427), we have — 

Log. sin. A (^ 18°) = 9.4899824 
^' 93.11 = 1.96899/O 

1.4589794 
' 43 = 1-6334685 

'' sin. ^(=42') = 9.8255109 

By reference to the log. tables, the last line of figures, as 
above, is found to be the sine of 42° ; therefore, the required 
angled is 42°. Then 180° - (18^ + 42°)= 120° = the angle <;. 

With these angles, or with any two of them, the third 
side c may be found by rules given in Art. 477. 





M^-- 






«/ 


R 


\1 


D 


N 


r 


^ 


k<- 


'A 


^ 


p 


V 


/ 


J 



Fig. 319. 

479. — Otoliqiie-Anijled TriiingJes : Sum and Difference 
of T%vo Anglc§. — Preliminary to a consideration of prob- 
lems in the third class of triangles, it is requisite to show the 
relation between the siii?i and difference of two angles. 

In Fig. 319, let the angle A JM and the angle A JN be 
the two given angles ; and let A J M be called angle ^4, and 
A J N, angle B. Now the sum and difference of the angles 
may be ascertained by the use of the sum and difference of 
the sines of the angles, and by the sum and difference of the 
tangents. In the diagram, in which the radius A J equals 



524 TRIGONOMETRY. 

unity, we have MP, the sine of angle A {— A J M), and 
NQ^ RP,thQ sine of angle B{= A J N). Then— 

IMP- RP=MR 

equals the difference of the sines of the angles ; and since 
PM' = PM— 

PM'+RP= RM\ 

equals the sum of the sines of the angles. 

With the radius yC' describe the arc JDE, and tangent 
to this arc draw FH parallel with MM', or perpendicular 
to^^. 

Then FD is the tangent of the angle M C N, and D H is 
the tangent of the angle N CM'. 

Now since an angle at the circumference is equal to half 
the angle at the centre standing on the same arc {Art. 355), 
therefore the measure of the angle M C N is the half oiMN, 
equals — 

^{AM—AN)=^{A-B). 

Similarly, we have — 

^{AM'+AN) = ^{A +B\ 

for the angle TV C: J/'. 

Therefore we have for the tangent of the angle M C N— 

FD = i?in. i{A -B), 

and, for the tangent of the angle N C M' — 

DH = tan. \{A + B). 

And, because FC D and M C R are homologous triangles, as, 
also, D CH and R CM\ therefore— 

M' R : MR : : DH : D F, 



SUM AND DIFFERENCE OF TWO ANGLES. 525 

sin. ^ + sin. B : sin. A — sin. B : : tan. i{A + B) : tan. ^{A — B), 

from which we have — 

sin. A — sin. B __ tan. ^ (A — B) ,^ 

sin. A + sin. B ~ tan. i {A + B)' ^ '^ 

To obtain a proper substitute for the first member of this 
expression we have, equation (195.) — 

sin. C 

c — a — , 

sm.yi 

or — 

c sin. A = a sin. C. (M.) 

We also have, equation (196.) — 

7 sin. C 
sm. B 
or — 

c sin. B — b sin. C. (N.) 

These two equations, (M.) and (N.), added, give — 

c sin. A ^ c sin. B = a sin. C -\- b sin. C. 
or — 

c (sin. A + sin. B) = sin. C(a -{- b). (P.) 

But, if equation (N.) be subtracted from equation (M.), we 
have — 

c sin. A — c sin. B = a sin. C — b sin. Cy 
or — 

c{?,in. A — sin. .5) =r ^ sin. C {a — b). (R.) 

If equation (R.) be divided by equation (P.), we have — 

<:(sin. A — sin. B) _ sin. C {a — b) 
c{s\n. A + sin. B) ~~ sin. C {a + b)* 



526 TRIGONOMETRY, 

which reduces to — 

sin. A — sin. B a 



sin. A + sin. B a + b 

The first member of this equation is identical with the first 
member of the above equation (D.), and therefore its equal, 
the second member, may be substituted for it ; thus — 

a — b _ t an. ^{A - B) 
'cT+l? ~ tan. i(yf + B)" 

From which we have — 

tan. 1 (^ - ^) z= tan. 1{A + B) ""-—^ . ' (203.) 

We have {Art. 431) the proposition, that if half the differ- 
ence of two quantities be subtracted from half their sum, the 
remainder will equal the smaller quantity. For example : 
if A represent the larger quantity and B the smaller, then — 

i{A+B)-i(A-B)^B; (204.) 

and, again, we also have {Art. 431) — 

■h{A +B) + ^{A-B)= A. (205.) 

480.— Oblique-Angled Triaiigle§: Third Cla§s.^The 
t/izrd class of problems comprises all those cases in which two 
sides of a triangle and their included angle are given, to 
find the other side and angles. 

In this case, as in the problems of the second class, the 
only requirement here is to find a second angle ; for then 
the problem becomes one belonging to the first class. But 
the finding of the second angle, in problems of the third 
class, is attended with more computation than it is in prob- 
lems of the second class. The process is as follows : Hav- 
ing one angle of a triangle, the sn^/i of the two remaining 



OBLIQUE-ANGLED TRIANGLES. 527 

angles is obtained by subtracting the given angle from 
180° — the sum of the three angles. 

Then with equation (203.) the difference of the two angles 
is obtained. And then, having the sum and dffcrence of the 
two angles, either may be found by one of the equations 
(204.) and (205.). 

For example : let Fig. 320 represent the triangle in 
which a {— 36 feet) and d{= 27 feet) are the given sides ; and 




C {= 105°) the angle included between the given sides, a and 
d. The sum of the two angles A and B, therefore, will be — 

{A+B) = 180- 105 =^75°, 

and the half of the sum of A and B \s ^- = 37° 30'. 

The sum of the given sides is 36 + 27 = 63, and their dif- 
ference is 36 — 27 = 9. 

Then from equation (203.) we have — 

tan. i{A-B)=^ tan. 37° 30^/. 

Solving this by logs. {Art. 427), we have — 

Log. tan. 37° 30' = 9.8849805 
9 = 0-9542425 

0.8392230 ■■ 

63 = 1.7993405 

tan. ^{A - i?)(=6° 15' 20- 5'') = 9-0398825 
Thus half the difference of A and ^ is 6° 15' 20- 5'', nearly. 



528 TRIGONOMETRY. 

By equation (204.) — 

37° 30' 
6° 15' 20.5^ 



The difference, 31° 14' 39*5'' = ^, 
and by equation (205.) — 

37-30 
6.15.20.5 



The sum, 43.45.20-5 = A 

From above, 31.14.39. 5 = B 

The given angle, 105. 0.0 = C 

The three angles, 180. o. o 

Thus, by adding together the three angles, the work is 
tested and proved. 

Having the three angles, the third side may now be 
found by the rule for problems of the first class. 

481. — Oblique-Anglcd Triangles : Fourth Class. — The 

fourth class comprises those problems in which the three 
sides of the triangle are given, to find the three angles. 

The method by which the problems of the fourth class 
are solved is to divide the triangle into two right-angled 
triangles; then, by the use of equation (129.), to find one 
side of one of these triangles, and then with this side to find 
one of the angles, then by rules for the second class prob- 
lems, obtain the second and third angles. 

Thus, from equation (129.), we have-^ 

^ 2C 

By the relation of sines to sides {Art. 476), we have (Fig. 

321)- 

b \ g '. \ sin. E : sin. F. 



TRIANGLES — FOURTH CLASS. 529 

But the angle ^5" is a right angle, of which the sine is unity, 
therefore — 

b '. g \ '. I : sin. F —J. 



Substituting for g its value as above, we have — 

sin./-^^'-(^±i) (^-^) . (206.) 

2 be 

To illustrate: let a, b, c {Fig. 321) be the three given sides 




of the triangle ABC, respectively equal to 12, 8 and 16 feet. 
With these, equation (206.) becomes — 

. ^ 16' — (12 + 8)(i2 - 8) 

sin. F = ^—^^ -* 

2 X 8 X 16 



„ 256 — (20 X 4) 
sin. F — — 



256 



sin. F — 



256 

Solving this by logarithms {Art, 427), we have- 
Log. 176 = 2.2455127 
'' 256 = 2.4082400 



Log. sin. 43° 26' = 9.8372727 
or, the angle at F equals 43° 26', nearly. Of the triangle 



530 



TRIGONOMETRY. 



A C E {Fig. 321), ^ is a right angle, therefore the sum of F 
and A, the two remaining angles, equals 90° {Art. 346). 
Hence, for the angle at y^, we have — 

^=90° -43° 26^ = 46° 34'. 

We now have two sides a and b and A, an angle opposite 
to one of them, to find B, a second angle. For this, equa- 
tion (199.) is appropriate. Thus — 



sin. B — sin. A - 
a 



This may be solved as shown in Art. 478. 

And, when the second angle is obtained, the third angle 
is found by subtracting the sum of the first and second an- 
gles from 180°. 

But to test the accuracy of the work, it is well to cojh- 
pute the angle C from the angle A, and the sides a and c. 
For this, equation (201.) will be appropriate. 

482. — Trigonometric Formulae : Rig^ht-Aiig^led Trian- 
gles. — For facility of reference the formulce of previous 




articles are here presented in tabular form. The symbols 
referred to are those of Fig. 322. 



FORMULAE IN TABULAR FORM. 



531 



Right-Angled Triangles. 



Given. 


Required. 


Formula. 


a, b, 

a, c, 

b, c, 


b, 
a. 




c ^ Va'+b\ 


b — V{c + a) {c — a). 


a = \/{c -^b){c- b). 


A, 
B, 


B, 

A. 


B= go"" -A. 
^ = 90" - B. 


B, a, 


b, 


b — a tan. B, 

a 
c — 


COS. B ' 


B,b, 


a, 


b 


b 
^ ~ sin. B ' 


B,c, 


a, 


a — c COS. B. 
b — c sin. B. 



483. — Trigonometrical Formulae: First Cia§§, Oblique. 

C 




• — The symbols of the formulae of the following table indi 
cate quantities represented in Fig. 323 by like symbols. 



532 



TRIGONOMETRY. 



Oblique-Angled Triangles: First Class. 



1 

Given. I Required. 


1 

Formula. 


A,B, 
A,C, 
B, C, 


c, 

A. 




C = i^o - A + B. 


B = i%o- A-\- C. 


A= i^o - B + C. 


A, B, b, 
A, C, c. 


a, 


J sin. A 1 
a = b-. — --. 1 
sin. B 

sin. A 

a = c -. --. 

sin. C 


A, B, a, 

B, C, c, 


b, 
b, 


J sin. B 

b — a -. . 

sin. A 

J sin. B 

b = C-. --. 

sm. 6 


A, C, a, 

B, C, b, 


Cy 


sin. C 
c — a -. — - - . 
sin. A 

_ 7 sin. C 


sin. B 



484. — Trigonometrical Formulae : Second €la§s, Oblique. 

— The symbols in the formulae of the following table refer 
to quantities represented in Fig. 323, by like symbols. 



J 



FORMUL/E FOR TRIANGLES, SECOND CLASS. $33 



Oblique-Angled Triangles: Second Class. 



Given. 


Required. 


FORMUL.^. 


B, a, b, 

i 


A, 
A, 


sin. A — sin. B -. 

b 

Sin. A — sin. C -. 


A, a, b, 
C, b, c, 


B, 
B, 


sin. B — sin. A -. 
a 

Sin. B = sin. C -. 
c 


A, a, c, 

B. b, c, 




sin. C = sin. A -. 
a 

sin. C — sin. B-. 
b 


B, C, 

A,C, 


A, 
B, 




A = iSo - B -{- C. 


B = iSo-A + C. 


A,B, 


C = iSo- A +B. 


For — 


a, 


See Formulae, First Class. 



534 



TKIGONOMETRY. 



4-86. — Trigonometrical Formulae : Third €las$«, Oblique. 

— The symbols in the formulse of the following table refer 
to quantities shown by like symbols in Fig, 323. 



Oblique- Angled Triangles: Third Class. 



Given, 


Required. 


Formula. 




A+B, 


^ +^= 180- C 


C, a, b, 


A-B, 


tan. iM - ^) =: tan. ^{A-^ B)^~^. 

a -V b 




A, 


A=\{A-\-B)-r \{A -B). 




B, 


B =i{A-\-B)-^{A -B). 




C +B, 


C + B = 1^0 - A. 


A, b, c, 


C-B, 


tan. 1{C ^)-tan. i((f +i?)^""^. 




c, 


C = ^{C + B) +i{C-B). 




B, 


B = ^{C-^B)-i{C-B). 




C +A. 


C -{- A = iSo - B. 


B, a, c, 


C-A, 


tan. 1{C A) = i^n.i{C+A)'~'^. 




c, 


C=i{C + A) + h{C-A), 




A, 


A=i{C-hA) + i(C-A). 



For the remaining side consult formulas for the Jirst 
class. 



486. — Trigonometrical Formulae : Fourtli Cla§s, Ob- 
lique. — The symbols in the formulae of the following table 
refer to quantities shown by like symbols in Fzg. 321. 



formula for triangles, fourth class. 535 

Oblique-Angled Triangles: Fourth Class. 



Given a, b^ c, to find A,B, C. 




2bc 




A = go - F. 




sin. B = sin. A -. 
a 




sin. C — sin. A -. 
a 




C = \Zo-yA + B). 



SECTION XV.— DRAWING. 

487. — General Remarks. — A knowledge of the proper- 
ties and principles of lijies can best be acquired by practice. 
Although the various diagrams throughout this work may 
be understood by inspection, yet they will be impressed 
upon the mind with much greater force, if they are actually 
drawn out with pencil and paper by the student. Science 
is acquired by study — art by practice ; he, therefore, who 
would have anything more than a theoretical (which must 
of necessity be a superficial) knowledge of carpentry and 
geometry, will provide himself with the articles here speci- 
fied, and perform all the operations described in the fore- 
going and following pages. Many of the problems may 
appear, at the first reading, somewhat confused and intricate ; 
but by making one line at a time, according to the explana- 
tions, the student will not only succeed in copying the fig- 
ures correctly, but by ordinary attention will learn the 
principles upon which they are based, and thus be able to 
make them available in any unexpected case to which they 
may apply. 

488. — Articles Required. — The following articles are 
necessary for drawing, viz. : a drawing-board, paper, draw- 
ing-pins or mouth-glue, a sponge, a T-square, a set-square, 
two straight-edges, or flat rulers, a lead pencil, a piece of 
india-rubber, a cake of india-ink, a set of drawing-instru- 
ments, and a scale of equal parts. 

489. — The I>raxvicig-Board. — The size of the drawing- 
board must be regulated according to the size of the draw- 
ings which are to be made upon it. Yet for ordinary prac- 
tice, in learning to draw, a board about fifteen by twenty 
inches, and one inch thick, will be found large enough, and 



DRAWING PAPER. 53/ 

more convenient than a larger one. This board should be 
well seasoned, perfectly square at the corners, and without 
clamps on the ends. A board is better without clamps, 
because the little service they are supposed to render by 
preventing the board from warping is overbalanced by the 
consideration that the shrinking of the panel leaves the 
ends of the clamps projecting beyond the edge of the board, 
and thus interfering with the proper working of the stock 
of the T-square. When the stuff is well-seasoned, the warp- 
ing of the board will be but trifling ; and by exposing the 
rounding side to the fire, or to the sun, it may be brought 
back to its proper shape. 

490. — Drawing-Paper. — For mere line drawings, it is 
unnecessary to use the best drawing-paper ; and since, where 
much is used, the expense will be considerable, it is desirable 
for economy to procure a paper of as low a price as will be 
suitable for the purpose. The best paper is made in Eng- 
land and water-marked " Whatman." This is a hand-made 
paper. There is also a machine-made paper at about half- 
price, and the manilla paper, of various tints of russet color, 
is still less in price. These papers are of the various sizes 
needed, and are quite sufficient for ordinary drawings. 

491. — To Secure tlie Paper to tlie :Board. — A drawing- 
pin is a small brass button, having a steel pin projecting from 
the underside. By having one of these at each corner, the 
paper can be fixed to the board ; but this can be done in a 
better manner with mouth-ghie. The pins will prevent the 
paper from changing its position on the board ; but, more 
than this, the glue keeps the paper perfectly tight and 
smooth, thus making it so much the more pleasant to work 
on. 

To attach the paper with mouth-glue, lay it with the 
bottom side up, on the board ; and with a straight-edge and 
penknife cut off the rough and uneven edge. With a 
sponge moderately wet rub all the surface of the paper, 
except a strip around the edge about half an inch wide. As 
soon as the glistening of the water disappears turn the sheet 



533 DRAWING. 

over and place it upon the board just where you wish it 
glued. Commence upon one of the longest sides, and pro- 
ceed thus : lay a flat ruler upon the paper, parallel to the 
edge, and within a quarter of an inch of it. With a knife, 
or anything similar, turn up the edge of the paper against 
the edge of the ruler, and put one end of the cake of mouth- 
glue between your lips to dampen it. Then holding it up- 
right, rub it against and along the entire edge of the paper 
that is turned up against the ruler, bearing moderately 
against the edge of the ruler, Avhich must be held firmly 
with the left hand. Moisten the glue as often as it becomes 
dry, until a sufficiency of it is rubbed on the edge of the 
paper. Take away the ruler, restore the turned-up edge to 
the level of the board, and lay upon it a strip of pretty stiff 
paper. By rubbing upon this, not very hard but pretty 
rapidly, with the thumb-nail of the right hand, so as to cause 
a gentle friction and heat to be imparted to the glue that is 
on the edge of the paper, you will make it adhere to the 
board. The other edges in succession must be treated in 
the same manner. 

Some short distances along one or more of the edges 
may afterward be found loose ; if so, the glue must again 
be applied, and the paper rubbed until it adheres. The 
board must then be laid away in a warm or dry place ; and 
in a short time the surface of the paper will be drawn out, 
perfectly tight and smooth, and ready for use. The paper 
dries best when the board is laid level. When the drawing 
is finished lay a straight-edge upon the paper and cut it 
from the board, leaving the glued strip still attached. This 
may afterward be taken off by wetting it freely with the 
sponge, which will soak the glue and loosen the paper. Do 
this as soon as the drawing is taken off, in order that the 
board may be dry when it is wanted for use again. Care 
must be taken that, in applying the glue, the edge of the 
paper does not become damper than the rest ; if it should, 
the paper nmst be laid aside to dry (to use at another time) 
and another sheet be used in its place. 

Sometimes, especially when the drawing-board is new, 
the paper will not stick very readily ; but by persevering 



THE T-SQUARE. 



539 



this difficulty may be overcome. In the place of the mouth- 
glue a strong solution of gum-arabic may be used, and on 
some accounts is to be preferred ; for the edges of the paper 
need not be kept dry, and it adheres more readily. Dissolve 
the gum in a sufficiency of warm water to make it of the 
consistency of linseed-oil. It must be apphed to the paper 
with a brush, when the edge is turned up against the ruler, 
as was described for the mouth-glue. If two drawing-boards 
are used, one may be in use while the other is laid away to 
dry ; and as they may be cheaply made, it is advisable to 
have two. The drawing-board having a frame around it, 
commonly called a panel board, may afford rather more 
facility in attaching the paper when this is of the size to 




Fig. 324. 

suit ; yet it has objections which overbalance that consid- 
eration. 

492. — The T-Square. — A T-squarc of mahogany, at once 
simple in its construction and affording all necessary service, 
may be thus made : let the stock or handle be seven inches 
long, two and a quarter inches wide, and three eighths of an 
inch thick ; the blade, twenty inches long (exclusive of the 
stock), two inches wide, and one eighth of an inch thick. In 
joining the blade to the stock, a very firm and simple joint 
may be made by dovetailing it — as shown at Fig. 324. 

493. — The Set-Square. — The set-square is in the form of 
a right-angled triangle ; and is commonly made of mahogany, 



540 DRAWING. 

one eighth of an inch in thickness. The size that is most 
convenient for general use is six inches and three inches 
respectively for the sides which contain the right angle, 
although a particular length for the sides is by no means 
necessary. Care should be taken to have the square corner 
exactly true. This, as also the T-square and rulers, should 
have a hole bored through them, by which to hang them 
upon a nail when not in use. 

494. — The Rulers. — One of the rulers may be about 
twenty inches long, and the other six inches. The pencil 
ought to be hard enough to retain a fine point, and yet not 
so hard as to leave ineffaceable marks. It should be used 
lightly, so that the extra marks that are not needed when 
the drawing is inked, may be easily rubbed off with the 
rubber. The best kind of india-ink is that which wall easily 
rub off upon the plate ; and, w^hen the cake is rubbed against 
the teeth, will be free from grit. 

495. — The Instruments. — The draiving-instriimeiits may 
be purchased of mathematical instrument makers at various 
prices ; from one to one hundred dollars a set. In choosing 
a set, remember that the lowest price articles are not always 
the cheapest. A set, comprising a. sufficient number of 
instruments for ordinary use, well made and fitted in a ma- 
hogany box, may be purchased of the mathematical instru- 
ment makers in New York for four or five dollars. But for 
permanent use those which come at ten or twelve dollars 
will be found to be better. 

496. — The ScaBe of Equal Parts. — The best scale of 
equal parts for carpenters' use, is one that has one eighth, 
three sixteenths, one fourth, three eighths, one half, five 
eighths, three fourths, and seven eighths of an inch, and one 
inch, severally divided into tivelftJis, instead of being divided, 
as they usually are, into tenths. By this, if it be required 
to proportion a drawing so that every foot of the object 
represented will upon the paper measure one fourth of an 
inch, use that part of the scale which is divided into one 



THE SET-SQUARE. 54 1 

fourths of an inch, taking for every foot one of those divis- 
ions, and for every inch one of the subdivisions into twelfths ; 
and proceed in hke manner in proportioning a drawing to 
any of the other divisions of the scale. An instrument in 
the form of a semi-circle, called a protractor, and used for 
laying down and measuring angles, is of much service to 
surveyors, and occasionally to carpenters. 

4-97. — The Use of tlie Set-Square. — In drawing parallel 
lines, when they are to be parallel to either side of the 
board, use the T-square ; but when it is required to draw 
lines parallel to a line which is drawn in a direction oblique 




Fig. 325. 

to either side of the board, the set-square must be used. 
Let ab {Fig. 325) be a line, parallel to which it is desired to 
draw one or more lines. Place any edge, as cd, of the set- 
square even with said line ; then place the ruler gh against 
one of the other sides, as ce, and hold it firmly ; slide the 
set-square along the edge of the ruler as far as it is desired, 
as at /; and a line draw^n by the edge if will be parallel 
to a b. 

To draw a line, as kl {Fig. 326), perpendicular to another, 
as a bj set the shortest edge of the set-square at the line a b ; 
place the ruler against the longest side (the hypothenuse of 
the right-angled triangle) ; hold the ruler firmly, and slide 
the set-square along until the side cd touches the point k] 
then the line Ik^ drawn by it, will be perpendicular to ab. 



542 DRAWING. 

In like manner, the drawing of other problems may be facil- 
itated, as will be discovered in using the instruments. 

498. — Directions for Drawing. — In drawing a problem, 
proceed, with the pencil sharpened to a point, to lay down 
the several lines until the Avhole figure is completed, ob- 
serving to let the lines cross each other at the several angles, 
instead of merely meeting. By this, the length of every 
Hne will be clearly defined. With a drop or two of water, 
rub one end of the cake of ink upon a plate or saucer, until 
a sufficiency adheres to it. Be careful to dry the cake of 




Fig 326. 

ink ; because if it is left wet it will crack and crumble in 
pieces. With an inferior camel's-hair pencil add a little 
water to the ink that w^as rubbed on the plate, and mix it 
well. It should be diluted sufficiently to flow freely from 
the pen, and yet be thick enough to make a black line. With 
the hair pencil place a little of the ink between the nibs of 
the drawing-pen, and screw the nibs together until the pen 
makes a fine line. Beginning Avith the curved lines, proceed 
to ink all the lines of the figure, being careful now to make 
every line of its requisite length. If they are a trifle too 
short or too long the drawing will have a ragged appear- 
ance ; and this is opposed to that neatness and accuracy 
which is indispensable to a good drawing. When the ink 
is dry efface the pencil-marks with the india-rubber. If the 



PUTTING THE DRAWING IN INK. 543 

pencil is used lightly they will all rub off, leaving those lines 
only that were inked. 

In problems all auxiliary lines are drawn light ; while the 
lines given and those sought, in order to be distinguished at 
a glance, are made much heavier. The heavy lines are 
made so by passing over them a second time, having the 
nibs of the pen separated far enough to make the lines as 
heavy as desired. If the heavy lines are made before the 
drawing is cleaned with the rubber they will not appear so 
black and neat, because the india-rubber takes away part 
of the ink. If the drawing is a ground-plan or elevation of 
a house, the shade-lines, as they are termed, should not be 
put in until the drawing is shaded ; as there is danger of the 
heavy lines spreading when the brush, in shading or color- 
ing, passes over them. If the lines are inked with common 
writing-ink they will, however fine they may be made, be 
subject to the same evil ; for which reason india-ink is the 
only kind to be used. 



SECTION XVI.— PRACTICAL GEOMETRY. 

4-99. — Definitions. — Geometry treats of the properties of 
magnitudes. 

A point has neither length, breadth, nor thickness. 

A line has length only. 

Superficies has length and breadth only. 

A plane is a surface, perfectly straight and even in every 
direction ; as the face of a panel when not warped nor 
winding. 

A solid has length, breadth, and thickness. 

A rights or straight, line is the shortest that can be drawn 
between two points. 

Parallel lines are equidistant throughout their length. 




Fig, 327. Fig. 328. Fig. 329. 

An angle is the inclination of two lines towards one an- 
other {Fig. 327). 

A right angle has one line perpendicular to the other 
{Fig 328). 

Kxi oblique angle is either greater or less than a right 
angle {Figs. 327 and 329). 

An acute angle is less than a right angle {Fig. 327). 

An obtuse angle is greater than a right angle {Fig. 329). 

When an angle is denoted by three letters, the middle 
one, in the order they stand, denotes the angular point, and 
the other two the sides containing the angle ; thus, let a, b, c 
{Fig. 327) be the angle, then b will be the angular point, and 
ab and be will be the two sides containing that angle. 



TRIANGLES AND RECTANGLES. 



545 



A triangle is a superficies having three sides and angles 
[Figs. 330, 331, 332, and 333). 

An equilateral triangle has its three sides equal {Fig. 330). 
An isosceles triangle has only two sides equal (Fig. 331). 





Fig. 330. 



Fig. 331, 



A scalene triangle has all its sides unequal {Fig. 332). 
A right-angled triangle has one right angle {Fig. 333). 
An acute-angled triangle has all its angles acute {Figs. 330 
and 331). 




Fig. 332. 



Fig. 333. 



An obtuse-angled triangle has one obtuse angle {Fig. 332). 
A quadrangle has four sides and four angles {Figs. 334 to 

339)- 

A parallelogram is a quadrangle having its opposite sides 
parallel {Figs. 334 to 337). 



Fig. 334. 



Fig. 335. 



being 



right 



A rectangle is a parallelogram, its angles 
angles {Figs. 334 and 335). 

A square is a rectangle having equal sides {Fig. 334). 

A rhombus is an equilateral parallelogram having oblique 
angles {Fig. 336). 



546 



PRACTICAL GEOMETRY. 



A rhomboid is a parallelogram having oblique angles 
{Fig. 337). 

A trapezoid is a quadrangle having only two of its sides 
parallel {Fig. 338). 



Fig. 336. 




7 



Fig. 337. 



A trapezium is a quadrangle which has no two of its sides 
parallel {Fig, 339). 

A polygon is a figure bounded by right lines. 

A regular polygo?t has its sides and angles equal. 

An irregular polygon has its sides and angles unequal. 




Fig. 338. 



Fig. 339. 



A trigon is a polygon of three sides {Figs. 330 to 333) ; a 
tetragon has four sides {Figs. 334 to 339) ; a pentagon has five 
{Fig. 340) ; a hexagon six (Z^^^. 341) ; a heptagon seven (F/^. 
342) ; an octagon eight (i^^^. 343) ; a nonagon nine ; a decagon 
ten ; an undecagon eleven ; and a dodecagon twelve sides. 




Fig. 340. 



Fig. 341. 



Fig. 342. 



Fig. 343. 



A circle is a figure bounded by a curved line, called the 
circumference, which is everywhere equidistant from a cer- 
tain point within, called its centre. 

The circumference is also called the periphery, and some- 
times the circle. 



PARTS OF THE CIRCLE. 



547 



The radius of a circle is a right line drawn from the 
centre to any point in the circumference {ab^ Fig. 334). 

All the radii of a circle are equal. 

The diameter is a right line passing through the centre, 
and terminating at two opposite points in the circumference. 
Hence it is twice the length of the radius {cd^ Fig. 344.) 




Fig. 344. 

An arc of a circle is a part of the circumference {c b, or 
bed, Fig. 344). 

A chord is a right line joining the extremities of an arc 

{b d, Fig. zAA)- ^ 

A segment is any part of a circle bounded by an arc and 
its chord {A, Fig. 344). 





Fig. 345. 



A sector is any part of a circle bounded by an arc and 
two radii, drawn to its extremities {B, Fig. 344). 

A quadrant, or quarter of a circle, is a sector having a 
quarter of the circumference for its arc [C, Fig. 344). 

A tangent is a right line which, in passing a curve, 
touches, without cutting it {f g, Fig. 344). 



548 



PRACTICAL GEOMETRY. 



A cone is a solid figure standing upon a circular base di- 
minishing in straight lines to a point at the top, called its 
vertex {Fig. 345). 

The axis of a cone is a right line passing through it, 
from the vertex to the centre of the circle at the base. 

An ellipsis is described if a cone be cut by a plane, not 
parallel to its base, passing quite through the curved surface 
{a b, Fig. 346). 

A parabola is described if a cone be cut by a plane, par- 
allel to a plane touching the curved surface {cd, Fig. 346 — 
cd being parallel to f g). 

An hyperbola is described if a cone be cut by a plane. 




Fi( 



347- 



parallel to any plane within the cone that passes through its 
vertex (eh, Fig. 346). 

Foci are the points at which the pins are placed in de- 
scribing an ellipse (see Art. 548, and /,/, Fig. 347). 

The transverse axis is the longest diameter of the ellipsis 
{a b, Fig. 347). 

The conjugate axis is the shortest diameter of the ellipsis ; 
and is, therefore, at right angles to the transverse axis {cd, 

J\?' 347). 

The parameter is a right line passing through the focus 
of an ellipsis, at right angles to the transverse axis, and ter- 
minated by the curve {gh and gt, Fig. 347). 



RIGHT LINES AND ANGLES. 



549 



A diameter of an ellipsis is any right line passing through 
the centre, and terminated by the curve [kl, or ;;/ ;/, Fig. 347). 

A diameter is conjugate to another when it is parallel to a 
tangent drawn at the extremity of that other — thus, the di- 
ameter inn {Fig. 347) being parallel to the tangent op, is 
therefore conjugate to the diameter kl. 

A double ordinate is any right line, crossing a diameter of 
an ellipsis, and drawn parallel to a tangent at the extremity 
of that diameter {i t, Fig. 347). 

A cylinder is a solid generated by the revolution of a 
right-angled parallelogram, or rectangle, about one of its 




Fig. 348. 



Fig. 349. 



sides ; and consequently the ends of the cylinder are equal 
circles {Fig. 348). 

The axis of a cylinder is a right line passing through it 
from the centres of the two circles which form the ends. 

A segjnent of a cylinder is comprehended under three 
planes, and the curved surface of the cylinder. Two of 
these are segments of circles ; the other plane is a parallelo- 
gram, called by way of distinction, the plane of the segment. 
The circular segments are called the ends of the cylinder 
{Fig. 349). 

PROBLEMS. 

RIGHT LINES AND ANGLES. 



500. — To Bisect a l.iiie. — Upon the ends of the line ab 
{Fig. 350) as centres, with any distance for radius greater 
than half ab, describe arcs cutting each other in c and d\ 



550 



PRACTICAL GEOMETRY 



draw the line cd, and the point e, where it cuts ab, will be 
the middle of the line ab. 

In practice, a line is generally divided with the com- 
passes, or dividers ; but this problem is useful where it is 




desired to draw, at the middle of another line, one at right 
angles to it. (See Art. 514.) 

601. — To Erect a Perpendicular. — From the point a 
{Fig. 351) set off any distance, as ab, and the same distance 
from ^ to ^ ; upon r, as a centre, with any distance for radius 
greater than ca, describe an arc at d\ upon b, with the same 




Fig. 351. 



radius, describe another at d\ join d and a, and the line da 
will be the perpendicular required. 

This, and. the three following problems, ai'e more easily 
performed by the use of the set-square (see Art. 493). Yet 
they are useful when the operation is so large that a set- 
square cannot be used. 



TO ERECT A PERPENDICULAR. 



551 



502. — To let Fall a Perpendicular. — Let a {Fig. 352) be 
the point above the line be from which the perpendicular is 
required to fall. Upon a^ with any radius greater than ad, 
describe an arc, cutting b c 2X e and /; upon the points e and 
/, with any radius greater than ed^ describe arcs, cutting 





c 


t 




h 


\ 


d 








r 


^f 



g 
Fig. 352. 

each other at g", join a and g, and the line ad wiW be the 
perpendicular required. 

603. — To Erect a Perpendicular at the End of a Line. 

— Let a {Fig. 353), at the end of the line c a, be the point at 
which the perpendicular is to be erected. Take any point, 
as b, above the line cay and with the radius ba- describe the 
arc dae\ through d and b draw the line de\ join e and «, 
then e a will be the perpendicular required. 




The principle here made use of is a very important one, 
and is applied in many other cases (see Art. 510, 3d, and Art. 
513. For proof of its correctness, see Art. 352). 

A second method. Let b (Fig. 354), at the end of the hne 
a b, be the point at which it is required to erect a perpendic- 
ular. Upon b, with any radius less than b a^ describe the arc 
ce d\ upon c, with the same radius, describe the small arc at^. 



552 



PRACTICAL GEOMETRY. 



and upon e, another at d ; upon e and d, with the same or any 
other radius greater than half e d, describe arcs intersecting 
at /; join /and b, and the line fb will be the perpendicular 
required. This method of erecting a perpendicular, and 
that of the following article, depend for accuracy upon the 




Fig. 354. 

fact that the side of a hexagon is equal to the radius of the 
circumscribing circle. 

A third method. Let b {Fig. 355) be the given point at 
which it is required to erect a perpendicular. Upon b, with 
any radius less than ba, describe the quadrant def; upon d, 
with the same radius, describe an arc at e, and upon e an- 
other at c\ through d and e draw dc, cutting the arc in c\ 
join c and b, then c b will be the perpendicular required. 




Fig. 355. 

This problem can be solved by the six, eight and ten rule, 
as it is called, which is founded upon the same principle as 
the problems at Arts, 536, 537, and is applied as follows: 
let ad {Fig. 353) equal eight, and ae, six; then, if de equals 
ten, the angle ead \s> 2i right angle. Because the square of 
six and that of eight, added together, equal the square of 



EQUAL ANGLES. 553 

ten, thus : 6 x 6 = 36, and 8 x 8 = 64 ; 36 + 64 = 100, and 
10 X 10 = 100. Any sizes, taken in the same proportion, as 
six, eight and ten, will produce the same effect; as 3, 4 and 
5, or 12, 16 and 20. (See Art. 536.) 

By the process shown at Fig. 353, the end of a board may 
be squared without a carpenters'-square. All that is neces- 
sary is a pair of compasses and a ruler. Let <;^ be the edge 
of the board, and a the point at which it is required to be 
squared. Take the point b as near as possible at an angle 
of forty-five degrees, or on a initre-\\nQ from a, and at about 
the middle of the board. This is not necessary to the work- 
ing of the problem, nor does it affect its accuracy, but the 
result is more easily obtained. Stretch the compasses from 
/; to Uy and then bring the leg at a around to d; draw a line 
from d, through b, out indefinitely ; take the distance db and 
place it from b to e -, join e and a ; then ea will be at right 
angles to ca. In squaring the foundation of a building, or 
laying out a garden, a rod and chalk-line may be used in- 
stead of compasses and ruler. 

504. — To let Fall a Perpendicular near the End of a 
Line. — Let e {Fig. 353) be the point above the line c a, from 
which the perpendicular is required to fall. From e draw 
any line, as ed, obliquely to the fine c a; bisect ^<^at ^ ; upon 
by with the radius be, describe the arc ead\ join ^ and a] 
then ea will be the perpendicular required. 

605. — To Make an Angle (a§ edf, Fig. 356) Equal to a 

OiTcn Angle (as b a c), — From the angular point «, with any 




Fig. 356. 

radius, describe the arc bc\ and with the same radius, on 
the line de, and from the point d, describe the arc/^; take 
the distance be, and upon g, describe the small arc at /; 



554 



PRACTICAL GEOMETRY. 



join / and d\ and the angle e df will be equal to the angle 
bac. 

If the given line upon which the angle is to be made is 
situated parallel to the similar line of the given angle, this 
may be performed more readily with the set-square. (See 
Art, 497.) 

606. — To Bisect an Angle. — Let abc {Fig. 357) be the 
angle to be bisected. Upon b^ with any radius, describe the 




Fig. 357. 

arc a c ; upon a and c, with a radius greater than half a c, 
describe arcs cutting each other at d\ join b and d\ and bd 
will bisect the angle abc^ as was required. 

This problem is frequently made use of in solving other 
problems ; it should therefore be well impressed upon the 
memory. 

507 — To Trisect a Right Angle.— Upon a {Fig. 358), 
with any radius, describe the arc b c ; upon b and Cy with the 




Fig. 358. 

same radius, describe arcs cutting the arc be 2it d and e; 
from d and e draw lines to a, and they will trisect the angle, 
as was required. 



TO DIVIDE A GIVEN LINE. 555 

The truth of this is made evident by the following oper- 
ation : divide a circle into quadrants ; also, take the radius 
in the dividers, and space off the circumference. This will 
divide the circumference into just six parts. A semi-circum- 
ference, therefore, is equal to three, and a quadrant to one 
and a half of those parts. The radius, therefore, is equal to 
two thirds of a quadrant ; and this is equal to a right angle. 

508. — Tliroug^li a Given Point, to Draw a Line Parallel 
to a Given L<ine. — Let a (Fig. 359) be the given point, and 



Fig. 359. 

be the given line. Upon any point, as d, in the line be, with 
the radius da, describe the arc ae\ upon a, with the same 
radius, describe the arc de\ make de equal to ae; through 
e and a draw the line ea, which will be the line required. 
This is upon the same principle as Art. 505. 

609- — To Divide a Given tine into any Number of 
Equal Parts. — Let a b {Fig. 360) be the given line, and 5 the 
number of parts. Draw ^^ at any angle to ab\ on <^r, from 




a, set off five equal parts of any length, as at i, 2, 3, 4 and e ; 
join c and b\ through the points i, 2, 3, and 4, draw le ,2f, 
3^ and 4//, parallel to ^<^; which will divide the line ab, as 
was required. 



556 



PRACTICAL GEOMETRY. 



The lines ab and ac are divided in the same proportion. 
(See Art, 542.) 



THE CIRCLE. 



6(0. — To Find the Centre of a Circle. — Draw any chord, 
as ^^ {Fig. 361), and bisect it with the perpendicular cd\ bi- 




sect ^^ with the line e f, as at ^; then g is the centre, as was 
required. 

A second method. Upon any two points in the circumfer- 
ence nearly opposite, as a and b {Fig. 362), describe arcs cut- 




ting each other at c and d\ take any other two points, as e 
and /, and describe arcs intersecting, as at g and h ; join g 
and h and c and d\ the intersection is the centre. 

This is upon the same principle as Art. 514. 

A third method. Draw any chord, tx.-s ab {Fig. 363), and 
from the point a draw ^ ^ at right angles to ab \ join c and 
b ; bisect c ^ at d — which will be the centre of the circle. 



A TANGENT AT A GIVEN POINT 



557 



If a circle be not too large for the purpose, its centre 
may very readily be ascertained by the help of a carpenters'- 
square, thus : apply the corner of the square to any point in 
the circumference, as at a ; by the edges of the square 
(which the lines ab and ac represent) draw lines cutting the 




Fig. 363. 

circle, as at b and c ; join b and c ; then if be is bisected, as at 
d, the point <^ will be the centre. (See Art. 352.) 

51 L — At a Oil en Point in a Circle to Draw a Tangent 
thereto. — Let a {Fig. 364) be the given point, and b the cen- 




FiG. 364. 

tre of the circle. Join a and b ; through the point a, and at 
right angles to a b, draw cd ] then ^ ^ is the tangent required. 



512. — The §anie, ivithont niakln^i^ nse of the Centre of 
the Circle. — Let a {Fig. 365) be the given point. From a set 
off any distance to b, and the same from b to c ] join a and 
c\ upon a, with ab for radius, describe the arc dbc\ make 
d b equal to be\ through a and d draw a line; this will be 
the tangent required. 



558 PRACTICAL GEOMETRY. 

The correctness of this method depends upon the fact 
that the angle formed by a chord and tangent is equal to any 
inscribed angle in the opposite segment of the circle {ArL 
358) ; ad being the chord, and dca the angle in the opposite 
segment of the circle. Now, the angles dad and dca are 
equal, because the angles dad and dac are, by construction. 




Fig. 365, 

equal; and the angles dac and dca are equal, because the 
triangle a dc is d.n isosceles triangle, having its two sides, a d 
and dcy by construction equal ; therefore the angles dad and 
dca are equal. 

513. — A Circle and a Tangent Given, to Find the Point 
of Contact. — From any point, as a {Fig: 366), in the tangent 




Fig. 366. 

d c, draw a line to the centre d ; bisect ad at ^ ; upon e, with 
the radius ea, describe the arc a/d; f is the point of con- 
tact required. 

If / and d were joined, the line would form right angles 
with the tangent d c, (See Art. 352.) 



A CIRCLE THROUGH GIVEN POINTS. 



559 



5 1 4-, — Throug^li any Three Points not in a Straight Line, 
to Draw a Circle. — Let a, b and c {Fig. 367) be the three 
given points. Upon a and b, with any radius greater than 
half a b, describe arcs intersecting at d and e ; upon b and r, 
with any radius greater than half b c, describe arcs intersect- 
ing at / and g\ through d and e draw a right line, also 




Fig. 367. 

another through / and g\ upon the intersection h, with the 
radius ha, describe the circle a be, and it will be the one re- 
quired. 

615. — Three Points not in a Straight L«ine being Oiven, 
to Find a Fourth that shall, with the Three, Lie in the 
Circumference of a Circle. — Let abc {Fig. 368) be the given 
points. Connect them with right lines, forming the triangle 




Fig. 368. 



acb\ bisect the angle cba {Art. 506) with the line bd\ also 
bisect c a'wL e, and erect ed perpendicular to ac, cutting bd 
in d ; then d is the fourth point required. 

A fifth point may be found, as at /, by assuming a, d 3,nd 
b, as the three given points, and proceeding as before. So, 



560 



PRACTICAL GEOMETRY. 



also, any number of points may be found simply by using 
any three already found. This problem will be serviceable 
in obtaining short pieces of very fiat sweeps. (See Art. 240.) 
The proof of the correctness of this method is found in 
the fact that equal chords subtend equal angles {Art. 357). 
Join d and c\ then since ae and ec are, by construction, 
equal, therefore the chords a d and dc are equal ; hence the 
angles they subtend, dba and dbc, are equal. So, like- 
wise, chords draAvn from a to /, and from / to d, are equal, 
and subtend the equal angles dbf and fba. Additional 
points beyo7td a or b may be obtained on the same principle. 
To obtain a point beyond a, on b, as a centre, describe with 
any radius the arc ion\ make on equal \.o oi\ through b and 
71 draw b g\ on ^ as centre and with af for radius, describe 
the arc, cutting ^^ at g, then g is the point sought. 

516, — To I>e§cribe a Segrmcnt of a Circle by a Set-Tri- 
angle. — Let a b {Fig. 369) be the chord, and c d the height 




Fig. 369. 

of the segment. Secure two straight-edges, or rulers, in the 
position ^^ and c f, by nailing them together at ^, and affixing 
a brace from ^ to /; put in pins at a and b ; move the angu- 
lar point c in the direction acb\ keeping the edges of the 
triangle hard against the pins a and b ; a pencil held at c 
will describe the arc acb. 

A curve described by this process is accurately circular, 
and is not a mere approximation to a circular arc, as some 
may suppose. This method produces a circular curve, be- 
cause all inscribed angles on one side of a chord-line are 
equal {Art. 356). To obtain the radius from a chord and its 
yersed sine, see Art. 444. 

If the angle formed by the rulers at <: be a right angle, 



TO FIND THE VERSED SINE, 561 

the segment described will be a semi-circle. This problem 
is useful in describing centres for brick arches, when they 
are required to be rather flat. Also, for the head hang- 
ing-stile of a window-frame, where a brick arch, instead of a 
stone lintel, is to be placed over it. 

517. — To Find the Radius of an Arc of a Circle when 
the Chord and Versed Sine are Oiven. — The radius is equal 
to the sum of the squares of half the chord and of the versed 
sine, divided by twice the versed sine. This is expressed, 

algebraically, thus : r = — , where r is the radius, c the 

chord, and v the versed sine {ArL 444.). 

Exainple.-^\\i a given arc of a circle a chord of 12 feet 
has the rise at the middle, or the versed sine, equal to 2 feet, 
what is the radius ? 

Half the chord equals 6, the square of 6 is, 6 x 6 = 36 
The square of the versed sine is, 2x2=4 

Their sum equals, 40 

Twice the versed sine equals 4, and 40 divided b}' 4 equals 
ID. Therefore the radius, in this case, is 10 feet. This 
result is shown in less space and more neatly by using the 
above algebraical formula. For the letters substituting 

C^Y " / 1 9\2 2 

their value, the formula r — - — =^-— becomes r — ^' ^ ^ , 

2V 2X2 

and performing the arithmetical operations here indicated 
equals — 

6'+2'_36 + 4_4o 

4 "~ 4 ~ "^ ~ ^^• 

5(8. — To Find the Versed Sine of an Arc of a Circle 
when the Radius and Chord are Oiven. — The versed sine 
is equal to the radius, less the square root of the difference 
of the squares of the radius and half chord ; expressed alge- 
braically thus: v = r — Vr ' - {t)\ where r is the radius, v 
the versed sine, and c the chord. (Equation (161.) reduced.) 



562 



PRACTICAL GEOMETRY. 



Example. — In an arc of a circle whose radius is 75 feet, 
what is the versed sine to a chord of 120 feet? By the table 
in the Appendix it will be seen that — 

The square of the radius, 75, equals . . 5625 
The square of half the chord, 60, equals . 3600 

The difference is 2025 

The square root of this is . . . -45 
This deducted from the radius ... 75 

The remainder is the versed sine, = 30 

This is expressed by the formula, thus — 



2/ = 75 - 1/75 ^_ (i|oy =: 75 - 1/5625 - 3600 ::= 75 - 45 rrr 30. 



519. — To Describe the Segment of a Circle by Intersec- 
tion of Lines. — Let ab {Fig. 370) be the chord, and cd the 



^ £r 1 



1 k f 




height of the segment. Through c draw e f parallel \,o ab\ 
draw bf at right angles to cb\ make ce equal to cf\ draw 
ag and bh at right angles io ab\ divide ce^ cf, da, db, ag, 
and bh, each into a like number of equal parts, as four; 
draw the lines i 1,22, etc., and from the points 0, 0, and 0, 
draw lines to c, at the intersection of these lines trace the 
curve, a cb, which will be the segment required. 

In very large work, or in laying out ornamental gar- 
dens, etc., this will be found useful ; and where the centre 
of the proposed arc of a circle is inaccessible it will be inval- 
uable. (To trace the curve, see note at Art. 550.) 

The lines ea, cd, and fb, would, were they extended, 
meet in a point, and that point would be in the opposite 
side of the circumference of the circle of which acb is a 



ORDINATES TO AN ARC. 



563 



segment. The lines i i, 2 2, 3 3, would likewise, if extended, 
meet in the same point. The line cd^ if extended to the op- 
posite side of the circle, would become a diameter. The line 
fb forms, by construction, a right angle with be, and hence 
the extension of fb would also form a right angle with be, 
on the opposite side of <^^; and this right angle would be 
the inscribed angle in the semi-circle ; and since this is re- 
quired to be a right angle {Art. 352), therefore the construc- 
tion thus far is correct, and it will be found likewise that at 
each point in the curve formed by the intersection of the 
radiating lines, these intersecting lines are at right angles. 

520. — Ordinates. — Points in the circumference of a 
circle may be obtained arithmetically, and positively accu- 
rate, by the calculation of oreimates, or the parallel lines o i, 





2 / d f 

Fig. 371. 



02, 03, 04 (i^/^. 371). These ordinates are drawn at right 
angles to the chord-line a b, and they may be drawn at any 
distance apart, either equally distant or unequally, and there 
may be as many of them as is desirable ; the more there are 
the more points in the curve will be obtained. If they are 
located in pairs, equally distant from the versed sine e d, 
calculation need be made only for those on one side of ed, 
as those on the opposite side will be of equal lengths, re- 
spectively ; for example: o i, on the left-hand side of ed, is 
equal to o i on the right-hand side, o 2 on the right equals 
o 2 on the left, and in like manner for the others. 

The length of any ordinate is equal to the square root 
of the difference of the squares of the radius and abscissa, 
less the difference between the radius and versed sine {Art. 
445). The abscissa being the distance from the foot of 
the versed sine to the foot of the ordinate. Algebraically, 



564 PRACTICAL GEOMETRY. 



/ = yr"^ — x"^ — (r — d), where t is put to represent the ordi- 
nate ; X, the abscissa ; d, the versed sine ; and r, the radius. 

Example. — An arc of a circle has its chord ab {Fig. 371) 
100 feet long, and its versed sine cd, ^ feet. It is required 
to ascertain the length of ordinates for a sufficient number 
of points through which to describe the curve. To this end 
it is requisite, first, to ascertain the radius. This is readily 

/«\2 2 

done in accordance with Art. 517. For becomes 

2,2 2 ^ 

~ — 252-5 = radius. Havins: the radius, the curve 

2x5 ^ :> ^ 

might at once be described without the ordinate points, but 
for the impracticability that usually occurs, in large, flat 
segments of the circle, of getting a location for the centre, 
the centre usually being inaccessible. The ordinates are, 
therefore, to be calculated. In Fig. 371 the ordinates are 
located equidistant, and are 10 feet apart. It will only 
be requisite, therefore, to calculate those on one side of 
the versed sine cd. For the first ordinate o i, the formula 
t — Vr"" — x^ — {r — b) becomes — 



/= 4/252.5'- 10'- (252.5 - 5). 



= 1^63756.25 — 100 — 247.5. 
= 252.3019-247.5. 
= 4 . 8019 = the first ordinate, o i . 

For the second — 



t = 4/252.5' -20' -(252.5 - 5). 

=z 251.7066 — 247.5. 

= 4. 2066 = the second ordinate, o 2. 
For the third— 



/ = 4/252.5'- 30'- 247.5. 
= 250.7115 -247.5. 
= 3.21 15 = the third ordinate, 03. 



TO DESCRIBE A TANGED CURVE. 565 

For the fourth — 



/ = 4/252.5' — 40' — 247.5. 

= 249.3115 - 247.5. 

= 1-8115= the fourth ordinate, o 4. 

The results here obtained are in feet and decimals of a 
foot. To reduce these to feet, inches, and eighths of an 
inch, proceed as at Reduction of Decimals in the Appendix. 
If the two-feet rule, used by carpenters and others, were 
decimally divided, there would be no necessity of this re- 
duction, and it is to be hoped that the rule will yet be thus 
divided, as such a reform would much lessen the labor of 
computations, and insure more accurate measurements. 

Versed sine ^ <^ = ft. 5 -o = ft. 5 -o inches. 
Ordinates 01= 4-8019 = 4.91 inches, nearly. 

*' 2=r 4.2066= 4-2^ inches, nearly. 

" 03= 3-2115= 3 • 2|- inches, nearly. 

" 04= 1. 81 15 = 1 .9I inches, nearly. 

521. — In a Given Angle, to Describe a Tanged Curve. 

— Let abc{Fig, 372) be the given angle, and i in the line a ^, 




Fig. 372. 

and 5 in the line be, the termination of the curve. Divide 
I b and b 5 into a like number of equal parts, as at i, 2, 3, 4, 
and 5 ; join i and i, 2 and 2, 3 and 3, etc. ; and a regular 
curve will be formed that will be tangical to the line a b, at 
the point i, and to ^^ at 5. 

This is of much use in stair-building, in easing the angles 
formed between the wall-string and the base of the hall, 
also between the front string and level facia, and in many 
other instances. The curve is not circular, but of the form 
of the parabola {Fig. 418) ; yet in large angles the difference 



$66 PRACTICAL GEOMETRY. 

is not perceptible. This problem can be applied to describ- 
ing the curve for door-heads, window-heads, etc., to rather 
better advantage th^n Art, 516. For instance, let ad {Fig. 
373) be the width of the opening, and c d the height of the 




arc. Extend c d, and make de equal to cd\ join a and e, 
also e and b ; and proceed as directed above. 

622, — To De§cribe a Circle within any Given Triangle, 
§0 that the Sides of the Triangle shall toe Tangieal. — 

Let abc {Fig. 374) be the given triangle. Bisect the angles 




Fig. 374. 

a and b according to Art. 506 ; upon d, the point of intersec- 
tion of the bisecting lines, with the radius d e, describe the 
required circle. 

523. — About a Given Circle, to Describe an Equilateral 
Triangle. — Let ad be {Fig. 375) be the given circle. Draw 
the diameter c d] upon d, with the radius of the given circle, 
describe the arc aeb\ join a and b\ draw /^at right angles 
to dc ; make fc and eg each equal to ^^; from /, through 
«, draw fh, also from g, through b, draw gh\ then fgh 
will be the triangle required. 

624-, — To Find a Right Liine nearly Equal to the Cir- 
cumference of a Circle. — Let abed {Fig. 376) be the given 



A RIGHT LINE EQUAL TO A CIRCUMFERENCE. 



567 



circle. Draw the diameter ac\ on this erect an equilateral 
triangle aec according to Art, 525 ; draw^/ parallel \.q ac\ 
extend ec to /, also ea to g\ then gf will be nearly the 




Fig. 375. 

length of the semi-circle adc\ and twice gf will nearly 
equal the circumference of the circle ab c d,2iS was required. 
Lines drawn from e, through any points in the circle, as 
<?, and 0, to /, p and /, will divide gf in the same way as 
the semi-circle adc is divided. So, any portion of a circle 
may be transferred to a straight line. This is a very useful 




problem, and should be well studied, as it is frequently 
used to solve problems on stairs, domes, etc. 

Another method. Let a bfc {Fig. 377) be the given circle. 
Draw the diameter a c ; from d, the centre, and at right an- 



568 



PRACTICAL GEOMETRY. 



gles to ac^ draw db\ join b and c\ bisect be at e\ from d, 
through e, draw df\ then ^/ added to three times the di- 
ameter, will equal the circumference of the circle sufficiently 
near for many uses. The result is a trifle too large. If the 




• Fig. 377. 

circumference found by this rule be divided by 648 • 22 the 
quotient will be the excess. Deduct this excess, and the 
remainder will be the true circumference. This problem is 
rather more curious than useful, as it is less labor to perform 
the operation arithmetically, simply multiplying the given 
diameter by 3- 1416, or, where a greater degree of accuracy 
is needed, by 3-1415926. (See Art. 446.) 

POLYGONS, ETC. 

525. — Upon a Given Line to Construct an Equilateral 
Triangle. — Let a b {Fig. 378) be the given line. Upon a and 




Fig. 378. 

b, with a b for radius, describe arcs, intersecting at c ; join a 
and r, also c and b ; then acb will be the triangle required. 



526. — To Describe an Equilateral Rectangle, or Square. 

Let ab (Fig. 379) be the length of a side of the proposed 



POLYGONS IN CIRCUMSCRIBING CIRCLES. 



569 



square. Upon a and b, with a b for radius, describe the arcs 
a d and b c ; bisect the arc ^^ in /; upon e, with e f for ra- 
dius, describe the arc cfd\ join a and c^ c and d, <^and b\ 
then ^<;<^<5 will be the square required. 




627. — Witliin a Given Circle, to In§eribe an Equilateral 
Triangle, Hexagon or Dodecagon. — Let abed (Fig. 380) be 
the given circle. Draw the diameter bd\ upon b, with the 
radius of the given circle, describe the arc ae c\ join a and r, 
also a and d, and c and d — and the triangle is completed. 
For the hexagon : from a^ also from c, through e, draw the 
lines af and c g\ join a and b^ b and c^ c and /", etc., and the 




Fig. 380. 

hexagon is completed. The dodecagon may be formed by 
bisecting the sides of the hexagon. 

Each side of a regular hexagon is exactly equal to the 
radius of the circle that circumscribes the figure. For the 
radius is equal to a chord of an arc of 60 degrees ; and, as 
every circle is supposed to be divided into 360 degrees, there 
is just 6 times 60, or 6 arcs of 60 degrees, in the whole cir- 
cumference. A line drawn from each angle of the hexagon 






570 



PRACTICAL GEOMETRY. 



to the centre (as in the figure) divides it into six equal, equi- 
lateral triangles. 

628. — Witliin a Square to Inscribe an Octagon.^ — Let 

abed {Fig. 381) be the given square. Draw the diagonals 




a d and b c ; upon a, b, c, and d, with a e for radius, describe 
arcs cutting the sides of the square at i, 2, 3, 4, 5, 6, 7, and 8 ; 
join I and 2, 3 and 4, 5 and 6, etc., and the figure is com- 
pleted. 

In order to eight-square a hand-rail, or any piece that is 




Fig. 382. 

to be afterwards rounded, draw the diagonals a d and b c 
upon the end of it, after it has been squared-up. Set a 
gauge to the distance a e and run it upon the whole length 
of the stuff, from each corner both ways. This will show 



BUTTRESSED OCTAGON. 5/1 

how much is to be chamfered off, in order to make the piece 
octagonal {Art. 354). 

529. — To Find the Side of a Buttressed Oetag^on. — Let 

ABCDE (Fig. 382) represent one quarter of an octagon 
structure, having a buttress HFGJ at each angle. The 
distance M H, between the buttresses, being given, as also 
F G, the width of a buttress ; to find H C or CJ/vi\ order to 
obtain B C, the side of the octagon. Let B C, 2i side of the 
octagon, be represented by b\ or I? C by id. Let MH = a ; 
or J D — ia\ and J C — x. 
Then we have — 

JD+yC=CD, 
ia -{- X = ib, 
a + 2 X = b. 

¥or FG^ui p- or LG = KJ = \p. 

Now E D IS the radius of an inscribed circle and, as per 

equation (140.), equals r = (V2 + i)-. 

Also, F C is the radius of a circumscribed circle, and, as 

per equation (141.), equals R = ^2 V2 + 4-. 

The two triangles, CJK and C ED^ are homologous; 
for the angles at C are common and the angles at K and D 
are right angles. Having thus two angles of one equal 
respectively to the two angles of the other, therefore {Art. 
345) the remaining angles must be equal. Hence, the sides 
of the triangles are proportionate, or — 

ED '. EC '.'. JK : CJ 

r : R'.'.\p : x=^\p-. 

The value of the side, as above, is — 

R 

^zn a -V 2x ^^ a + p — . 



572 PRACTICAL GEOMETRY. 

And taking the value of R and r, as above, we have- 






Substituting this for — , we have — 



t? = a + p — 3 . 

1^2+ I 

The numerical coefficient of / reduces to 1-0823923 or 
I -0824, nearly. 

Therefore we have — 

b = a-^ \'0Z2Afp. (207.) 

Or : The side of a buttressed octagon equals the distance be- 
tween the buttresses plus I -0824 times the width of the face of 
the buttress. 

For example : let there be an octagon building, which 
measures between the buttresses, as at M H, 18 feet, and the 
face of the buttresses, as FG, equals 3 feet ; what, in such a 
building, is the length of a side B C^ For this, using equa- 
tion (207.), we have — 

b ~ \^ -\- 1-0824 X 3 
= 18 + 3-2472 
= 21 .2472. 

Or : The side of the octagon B C equals 21 feet and nearly 3 
inches. 

630. — Within a Given Circle to Inscribe any Reg^uiar 
Polygon — Let abc2 (Figs. 383, 384, and 385) be given circles. 
Draw the diameter a c ; upon this erect an equilateral trian- 
gle aec^ according to^r/. 525 ; divide ac into as many equal 
parts as the polygon is to have sides, as at i, 2, 3, 4, etc.; 
from ^, through each even number, as 2, 4, 6, etc., draw lines 



TO DESCRIBE ANY REGULAR POLYGON. 



573 



cutting the circle in the points 2, 4, etc. ; from these points 
and at right angles to ac draw lines to the opposite part 
of the circle ; this will give the remaining points for the 
polygon, as d, /, etc. 

In forming a hexagon, the sides of the triangle erected 




Fig. 383. 



Fig. 384. 



Fig. 385. 



upon ac (as at Fig: 384) mark the points d and /. This 
method of locating the angles of a polygon is an approxima- 
tion sufficiently near for man)^ purposes ; it is based upon 
the like principle with the method of obtaining a right line 
nearly equal to a circle {Art. 524). The method shown at 
Art. 531 is accurate. 




Fig. 386. 



Fig. 387. 



Fig. 388 



531. — Upon a Given Line to Bcscribe any Regular 
Polygon. — Let ab {Figs. 386, 387, and 388) be given lines, 
equal to a side of the required figure. From b draw ^<: at 
right angles to ^ <^ ; upon a and b, with a b for radius, describe 
the arcs acd and feb\ divide ac into as many equal parts 



574 PRACTICAL GEOMETRY. 

as the polygon is to have sides, and extend those divisions 
from c towards d\ from the second point of division, count- 
ing from c towards a, as 3 {Fig, 386), 4 {Fig. 387), and 5 [Fig. 
388), draw a line to b ; take the distance from said point of 
division to a, and set it from b to e\ join e and a ; upon the 
intersection o with the radius oa, describe the circle afdb\ 
then radiating lines, drawn from b through the even numbers 
on the arc a d, will cut the circle at the several angles of the 
required figure. 

In the hexagon {Fig. 387), the divisions on the arc «^are 
not necessary ; for the point is at the intersection of the 
arcs ad and fby the points / and d are determined by the 
intersection of those arcs with the circle, and the points 
above g and h can be found by drawing lines from a and b 
through the centre 0. In polygons of a greater number of 
sides than the hexagon the intersection comes above the 
arcs; in such case, therefore, the lines ae and b^ {Fig. 388) 
have to be extended before they will intersect. This method 
of describing polygons is founded on correct principles, and 
is therefore accurate. In the circle equal arcs subtend 
equal angles {Arts. 357 and 515). Although this method is 
accurate, yet polygons may be described as accurately and 
more simply in the following manner. It will be observed 
that much of the process in this method is for the purpose 
of ascertaining the centre of a circle that will circumscribe 
the proposed polygon. By reference to the Table of Poly- 
gons in Art. 442 it will be seen how this centre may be ob- 
tained arithmetically. This is the rule : multiply the given 
side by the tabular radius for polygons of a like number of 
sides with the proposed figure, and the product will be the 
radius of the required circumscribing circle. Divide this 
circle into as many equal parts as the polygon is to have 
sides, connect the points of division by straight lines, and 
the figure is complete. For example : It is desired to de- 
scribe a polygon of 7 sides, and 20 inches a side. The tabu- 
lar radius is 1-15238. This multiplied by 20, the product, 
23-0476 is the required radius in inches. The Rules for 
the Reduction of Decimals, in the Appendix, show how to 
change decimals to the fractions of a foot or an inch. From 



EQUAL FIGURES. 



575 



this, 23-0476 is equal to 23 jV inches, nearly. It is not needed 
to take all the decimals in the table, three or four of them 
will give a result sufficiently near for all ordinary practice. 

532. — To Construct a Triangle whose Sides shall he 
severally Equal to Three Given iLines. — Let a^ b and c {Fig. 

389) be the given lines. Draw the line de and make it equal 




Fig. 389. 

c\ upon ^, with b for radius, describe an arc at /; upon d, 
with a for radius, describe an arc intersecting the other at/; 
join d and /, also / and e ; then dfe will be the triangle 
required. 

533. — To Construct a Figure Equal to a Given, Right- 
lined Figure. — Let abed (Fig. 390) be the given figure. 
Make e f {Fig. 391) equal to cd] upon /, w^th da for radius, 




Fig. 390. 



Fig. 39] 



describe an arc at^; upon c, with ca for radius, describe an 
arc intersecting the other at^; join ^and e; upon / and g, 
with db and ab for radius, describe arcs intersecting at h\ 
join g and h, also h and /; then Fig. 391 will every way 
equal Fig. 390. 

So, right-lined figures of any number of sides may be 
copied, by first dividing them into triangles, and then pro- 



576 



PRACTICAL GEOMETRY. 



ceeding as above. The shape of the floor of any room, or 
of any piece of land, etc., may be accurately laid out by this 
problem, at a scale upon paper ; and the contents in square 
feet be ascertained by the next. 

634. — To Make a Parallelogram equal to a Criven 
Triangle. — Let abc {Fig. 392) be the given triangle. From 
a draw ^ ^ at right angles to bc\ bisect ad'va e\ through e 



/ 



d 
Fig. 392. 

draw fg parallel to bc\ from b and c draw b f and <:^ par- 
allel to de\ then bfgc will be a parallelogram containing a 
surface exactly equal to that of the triangle abc. 

Unless the parallelogram is required to be a rectangle, 
the Hues bf and eg need not be drawn parallel to de. If a 
rhomboid is desired they may be drawn at an oblique angle, 
provided they be parallel to one another. To ascertain the 
area of a triangle, multiply the base <^<; by half the perpen- 



e "V 



s ~p 

^^ e 

.^ C 



d f 

Fig. 393. 



dicular height da. In doing this it matters not which side 
is taken for base. 



536. — A Parallelogram being Given, to Construct An- 
other Equal to it, and Having a Side Equal to a Given L.ine. 

— Let A {Fig. 393) be the given parallelogram, and B the 
given line. Produce the sides of the parallelogram, as at 



SQUARE EQUAL TO TWO OR MORE SQUARES. 



577 



a, b, c, and d'; make cd equal to B\ through <^ draw cf par- 
allel to ^^; through e draw the diagonal ca\ from a draw 
af parallel to ed\ then C will be equal to A. (See Art. 340.) 

636. — To make a Square Equal to two or more Oiven 
Squares. — Let A and B {Fig. 394) be two given squares. 



A 


a '^v^. 




B 



Fig. 394. 

Place them so as to form a right angle, as at ^ ; join b and c ; 
then the square C, formed upon the line be, will be equal in 
extent to the squares A and B added together. Again : if 
a b {Fig. 395) be equal to the side of a given square, ca, placed 
at right angles to a b, be the side of another given square, 




Fig. 395. 

and cd, placed at right angles to cb, be the side of a third 
given square, then the square A, formed upon the line db, 
will be equal to the three given squares. (See Art. 353.) 

The usefulness and importance of this problem are pro- 
verbial. To ascertain the length of braces and of rafters in 



578 PRACTICAL GEOMETRY. 

framing, the length of stair-strings, etc., are some of the pur- 
poses to which it may be appUed in carpentry. (See note 
to Art. 503.) If the lengths of any two sides of a right- 
angled triangle are known, that of the third can be ascer- 
tained. Because the square of the hypothenuse is equal to 
the united squares of the two sides that contain the right 
angle. 

(i.) — The two sides containing the right angle being 
known, to find the hypothenuse. 

Rule. — Square each given side, add the squares together, 
and from the product extract the square root ; this will be 
the answer. 

For instance, suppose it were required to tind the length 
of a rafter for a house, 34 feet wide — the ridge of the roof 
to be 9 feet high, above the level of the wall-plates. Then 
17 feet, half of the span, is one, and 9 feet, the height, is the 
other of the sides that contain the right angle. Proceed as 
directed by the rule : 

17 9 . 

17 9 

119 81 = square of 9. 

17 289 = square of 17. 

289 = square of 17. 370 Product. 

I ) 370 ( 19-235 + = square root of 370 ; equal 19 feet 2f in., 
I I nearly ; which would be the required 

2Q )^7o length of the rafter. 

9 261 
3 82).. 900 
_^ 764^ 
3843)13600 

3 11529 

38465) -207100 
192325 



TO FIND THE LENGTH OF A RAFTER. 579 

(By reference to the table of square roots in the Appen- 
dix, the root of almost any number may be found ready 
calculated ; also, to change the decimals of a foot to inches 
and parts, see Rliles for the Reduction of Decimals in the 
Appendix.) 

Again : suppose it be required, in a frame building, to 
hnd the length of a brace having a run of three feet each 
way from the point of the right angle. The length of the 
skies containing the right angle will be each 3 feet ; then, as 
before — 

3 

9 = square of one side. 
3 times 3 = 9 = square of the other side. 

1 8 Product : the square root of which is 4 • 2426+ ft., 
or 4 feet 2 inches and |- full. 

(2.) — The hypothenuse and one side being known, to find 
the other side. 

jRu/c\ — Subtract the square of the given side from the 
square of the hypothenuse, and the square root of the prod- 
uct will be the answer. 

Suppose it were required to ascertain the greatest per- 
pendicular height a roof of a given span may have, when 
pieces of timber of a given length are to be used as rafters. 
Let the span be 20 feet, and the rafters of 3 x 4 hemlock 
joist. These come about 13 feet long. The known hy- 
pothenuse, then, is 13 feet, and the known side, 10 feet — 
that being half the span of the building. 

13 
13 



39 
•13 

169 = square of hypothenuse. 
10 times 10 = 100 = square of the given side. 

6q Product : the square root of which is 



58o PRACTICAL GEOMETRY. 

S-3o66+ feet, or 8 feet 3 inches and | full. This will be the 
greatest perpendicular height, as required. Again : suppose 
that in a story of 8 feet, from floor to floor, a step-ladder is 
required, the strings of which are to be of plank 12 feet 
long, and it is desirable to know the greatest run such a 
length of string Avill afford. In this case, the two given 
sides are — hypothenuse 12, perpendicular 8 feet. 

12 times 12 = 144 = square of hypothenuse, 
8 times 8 = 64 = square of perpendicular. 

80 Product : the square root of which is 
8-9442+ feet, or 8 feet 11 inches and -^^ — the answer, as re- 
quired. 

Many other cases might be adduced to show the utility 
of this problem. A practical and ready method of ascer- 
taining the length of braces, rafters, etc., when not of a great 
length, is to apply a rule across the carpenters' -square. 
Suppose, for the length of a rafter, the base be 12 feet and 
the height 7. Apply the rule diagonally on the square, so 
that it touches 12 inches from the corner on one side, and 7 
inches from the corner on the other. The number of inches 
on the rule which are intercepted by the sides of the square, 
I3f, nearly, will be the length of the rafter in feet ; viz., 13 
feet and -J of a foot. If the dimensions are large, as 30 feet 
and 20, take the half of each on the sides of the square, viz., 
15 and ID inches; then the length in inches across will be 
one half the number of feet the rafter is long. This method 
is just as accurate as the preceding ; but when the length of 
a very long rafter is sought, it requires great care and pre- 
cision to ascertain the fractions. For the least variation on 
the square, or in the length taken on the rule, would make 
perhaps several inches difference in the length of the rafter. 
For shorter dimensions, however, the result will be true 
enough. 

537. — To Make a Circle Equal to two Given Circles. — 

Let yl and B {Fig. 396) be the given circles. In the right- 
angled triangle abc make ab equal to the diameter of the 



SliMILAR FIGURES. 



581 



circle B, and cb equal to the diameter of the circle A ; then 
the hypothenuse ac will be the diameter of a circle C, which 
will be equal in area to the two circles A and B, added 



together. 




Any polygonal figure, as A {Fig. 397), formed on the hy- 
pothenuse of a right-angled triangle, will be equal to two 
similar figures, "^^ as B and C, formed on the two legs of the 
triangle. 




Fig. 397. 

538. — To Construct a Square Equal to a Given Rect- 
angle. — Let A {Fig. 398) be the given rectangle. Extend 
the side ab and make be equal to <^^; bisect <^ ^ in /, and 
upon /, with the radius fa, describe the semi-circle agc\ 
extend eb till it cuts the curve in g\ then a square bgJid, 
formed on the line b g, will be equal in area to the rectan- 
gle yi. 



* Similar figures are such as have their several angles respectively equal, 
and their sides respectively proportionate. 



582 



PRACTICAL GEOMETRY. 



Another method. Let A {Fig. 399) be the given rectangle. 
Extend the side ab and make ad equal to a c ; bisect ad in 
e\ upon e, with the radius ea, describe the semi-circle afd\ 
extend gb till it cuts the curve in /; join a and /; then 




Fig. 398. 

the square B, formed on the line af, will be equal in area to 
the rectangle A. (See Arts. 352 and 353.) 

639- — To Form a l^quare Equal to a Given Triangle. — 

Let a b {Fig. 398) equal the base of the given triangle, and b e 



A / 



Fio. 399. 

equal half its perpendicular height (see Fig. 392) ; then pro- 
ceed as directed at Art. 538. 



, 540.— Two Right L<iMCs being Given, to Find a Third 
Proportional Thereto. — Let A and B {Fig. 400) be the given 
lines. Make a b equal to A ; from a draw ac <it any angle 



PROPORTIONATE DIVISIONS IN LINES. 583 

with ab\ make ac and ad each equal to B\ join c and b\ 
from d draw ^^ parallel \.o cb\ then ae will be the third 
proportional required. That is, a e bears the same propor- 
tion to B as B does to A. 




541. — Three Rig;lit Lines beings Given, to Find a Fourth 
Proportional Thereto. — Let A, B, and C {Fig- 401) be the 
given lines. Make ab equal to A ; from a draw ac 2it any 



A 


(^ 








S,^-" 


c 


a 


e 

Fig. 401. 


I 



angle with a b ; make a c equal to B and a e equal to C ; join 
c and b ; from e draw e f parallel to cb\ then a f will be the 
fourth proportional required. That is, af bears the same 
proportion to C ^.s B does to A. 

To apply this problem, suppose the two axes of a given 
ellipsis and the longer axis of a proposed ellipsis are given. 
Then, by this problem, the length of the shorter axis to the 
proposed ellipsis can be found ; so that it will bear the same 
proportion to the longer axis as the shorter of the given 
ellipsis does to its longer. (See also Art. 559.) 

542. — ^A Liine with Certain Divisions beings Given, to 
Divide Another, Longer or Shorter, Given Line in the 
Same Proportion. — Let A {Fig. 402) be the line to be di- 
vided, and B the line with its divisions. Make a b equal to 
B with all its divisions, as at i, 2, 3, etc.; from a draw a c dit 
any angle with ab ; make ac equal to A ; join c and b ; from 



584 



PRACTICAL GEOMETRY. 



the points i, 2, 3, etc., draw lines parallel to cd; then these 
will divide the line <^ ^ in the same proportion as B is divided 
— as was required. 

This problem will be found useful in proportioning the 



c 
a 1 2 3 4 5 i 



Fig. 402. 

members of a proposed cornice, in the same proportion as 
those of a given cornice of another size. (See Art. 321.) So 
of a pilaster, architrave, etc. 

643- — Between Two Given Right Lines, to Find a 
mean Proportional. — Let A and B {Fig, 403) be the given 
lines. On the line ac make ab equal to A and be equal to 
B\ bisect ac in e; upon e, with ea for radius, describe the 
semi-circle adc; at b erect ^ </ at right angles to ^ ^ ; then 




, Fig. 403. 

bd will be the mean proportional between A and B. That 
IS, ab is to bd ?iS> bd is to be. This is usually stated thus: 
ab : b d : : bd : be, and since the product of the means 
equals the product of the extremes, therefore, abxbe^^bd^- 
This is shown geometrically at Art. 538. 



CONIC SECTIONS. 

544. — Definitions. — If a cone, standing upon a base 
that is at right angles with its axis, be cut by a plane, per- 



AXIS AND BASE OF PARABOLA. 



585 



pendicular to its base and passing through its axis, the sec- 
tion will be an isosceles triangle (as a be, Fig. 404) ; and the 
base will be a semi-circle. If a cone be cut by a plane in the 
direction e f the section will be an ellipsis ; if in the direction 
ml, the section will be a parabola; and if in the direction 
TO, an hyperbola. (See Art. 499.) If the cutting planes be 
at right angles with the plane ab e, then — 

545. — To Find the Axes of the Ellipsis: bisect e f {Fig. 
404) in g\ through g draw Ji i parallel \.o ab\ bisect h i in 7; 




Fig. 404. 

upon y, withy// for radius, describe the semi-circle }iki\ 
from ^ draw gk 2iX. right angles to Jii\ then twice gk will 
be the conjugate axis and e f the transverse. 

546. — To Find the Axis and Base of the Parabola. — 

Let m I {Fig. 404), parallel to ae, be the direction of the cut- 
ting plane. From m draw m d at right angles to ab; then 
Im will be the axis and height, and nid an ordinate and half 
the base, as at Figs. 417, 418. 



547. — To Find the Height, Base, and Transverse Axis 
of an Hyperbola. — Let r {Fig. 404) be the direction of the 



586 



PRACTICAL GEOMETRY. 



cutting plane. Extend o r and a c till they meet at ;/ ; from 
draw op at right angles \.o ab\ then r o will be the height, 
;/ r the transverse axis, and op half the base ; as at Fig. 419. 

548. — Tlie Axes toeing Giieii, to Find the Foci, and to 
Describe an Ellipsis Vvitli a String. — Let ab {Fig. 405) and 
cdhQ the given axes. Upon c, with ^^ or be for radius, de- 
scribe the arc //; then / and /, the points at which the 
arc cuts the transverse axis, will be the foci. At / and / 
place two pins, and another at c \ tie a string about the three 
pins, so as to form the triangle ffc ; remove the pin from c 
and place a pencil in its stead ; keeping the string taut, 




move the pencil in the direction cga; it will then describe 
the required ellipsis. The lines fg and gf show the posi- 
tion of the string when the pencil arrives at g. 

This method, when performed correctly, is perfectly ac- 
curate ; but the string is liable to stretch, and is, therefore, 
not so good to use as the trammel. In making an ellipse by 
a string or twine, that kind should be used which has the 
least tendenc}^ to elasticity. For this reason, a cotton cord, 
such as chalk-lines are commonly made of, is not proper for 
the purpose ; a linen or flaxen cord is much better. 



54-9, — Tlie Axes toeing CJiven, to Describe an JCllipsis 
with a Trammel. — Let ab and cd {Fig. 406) be the given 
axes. Place the trammel so that a line passing through the 
centre ot the grooves would coincide with the axes ; make 



ELLIPSE BY TRAMMEL. 



587 



the distance from the pencil c to the nut/ equal to half cd\ 
also, from the pencil e to the nut g equal to half a b ; letting 
the pins under the nuts slide in the grooves, move the tram- 
mel eg in the direction cbd\ then the pencil at c will de- 
scribe the required ellipse. 

A trammel may be constructed thus: take two straight 
strips of board, and make a groove on their face, in the cen- 
tre of their width ; join them together, in the middle of their 
length, at right angles to one another ; as is seen at Fig. 406. 
A rod is then to be prepared, having two movable nuts 
made of wood, with a mortise through them of the size of 
the rod, and pins under them large enough to fill the 
grooves. Make a hole at one end of the rod, in which to 




Fig. 406. 



place a pencil. In the absence of a regular trammel a tem- 
porar}^ one may be made, which, for any short job, will an- 
swer every purpose. Fasten two straight-edges at right 
angles to one another. Lay them so as to coincide with the 
axes of the proposed ellipse, having the angular point at the 
centre. Then, in a rod having a hole for the pencil at one 
end, place two brad-awls at the distances described at Art. 
549. While the pencil is moved in the direction of the 
curve, keep the brad-awls hard against the straight-edges, 
as directed for using the trammel-rod, and one quarter of 
the ellipse will be drawn. Then, by shifting the straight- 
edges, the other three quarters in succession may be drawn. 
If the required ellipse be not too large, a carpenters'-square 
may be made use of, in place of the straight-edges. 

An improved method of constructing the trammel is as 



588 



PRACTICAL GEOMETRY 



follows : make the sides of the grooves bevelling from the 
face of the stuff, or dove-tailing instead of square. Prepare 
two slips of wood, each about two inches long, which shall 
be of a shape to just fill the groove when slipped in at the 
end. These, instead of pins, are to be attached one to each 
of the movable nuts with a screw, loose enough for the nut 
to move freely about the screw as an axis. The advantage 
of this contrivance is, in preventing the nuts from slipping 
out of their places during the operation of describing the 
curve. 



550. — To De§cribe an Ellipsis by Ordinate^. — Let ab 

and cd {Fig. 407) be given axes. With c e qv c d for radius 




/12 3a 



describe the quadrant f gJi ; divide /"//, ac^ and e b, each into 
a like number of equal parts, as at i, 2, and 3 ; through 
these points draw ordinates parallel \.o cd and fg\ take the 
distance i i and place it at i /, transfer 27 to 2 vi, and 3 /^ to 
3 ;/ ; through the points a, 11, in, I, and c, trace a curve, and 
the eUipsis will be completed. 

The greater the number of divisions on a, c, etc., in this 
and the following problem, the more points in the curve can 
be found, and the more accurate the curve can be traced. 
If pins are placed in the points n, in, /, etc., and a thin slip 
of wood bent around by them, the curve can be made quite 
correct. This method is mostly used in tracing face-moulds 
for stair hand -railing. 



55 (. — To Describe an Ellipsis by Intersection of Lines. 

■Let ab and cd {Fig. 408) be given axes. Through c, draw 



ELLIPSE BY INTERSECTION OF LINES. 



589 



fg parallel X.o ab\ from a and b draw af and h g at right 
angles to ab\ divide f a^ gb, ac\ and eb, each into a like 
number of equal parts, as at i, 2, 3, and o, 0, o\ from i, 2, 
and 3, draw lines to c ; through o, 0, and o, draw lines from d, 




intersecting those drawn to c ; then a curve, traced through 
the points i, i, i, will be that of an ellipsis. 

Where neither trammel nor string is at hand, this, per- 
haps, is the most ready method of drawing an ellipsis. The 
divisions should be small, where accuracy is desirable. By 
this method an ellipsis may be traced without the axes, pro- 
vided that a diameter and its conjugate be given. Thus, ab 




and cdiFig. 409) are conjugate diameters: fg\^ drawn par- 
allel to ab^ instead of being at right angles to r^; also, fa 
and gb are drawn parallel to c d, instead of being at right 
angles to a b. 






590 



PRACTICAL GEOxMETRY. 



552- — To Describe an Ellipsis by Intersecting Arcs» — 

Let a b and c d {Fig. 410) be given axes. Between one of the 
foci, / and /, and the centre r, mark any number of points, 
at random, as i, 2, and 3 ; upon / and /, with b i for radius, 
describe arcs at g, g, g, and g] upon / and /, with a i for 




radius, describe arcs intersecting- the others at g,g,g, and^; 
then these points of intersection will be in the curve of the 
ellipsis. The other points, Ji and /, are found in like manner, 
viz.: h is found by taking b2 for one radius, and a2 for the 
other ; i is found by taking b 3 for one radius, and a 3 for the 




Fig. 411. 

Other, always using the foci for centres. Then by tracing a 
curve through the points c, g, h, i, b, etc., the ellipse will be 
completed. 

This problem is founded upon the same principle as that 
of the string. This is obvious, when we reflect that the 
length of the string is equal to the transverse axis, added to 



TO DESCRIBE AN OVAL. 



591 



t\\e distance between the foci. See Fzo-, 405, in which c/ 
equals ae, the half of the transverse axis. 

553- — To I>e§cribc a Figure ;^early in the Shape of an 
Ellipsis, hy a Pair of Compasses. — Let a b and c d {Fig. 41 1) 
be given axes. From c draw c e parallel to ab \ from a draw 
ae parallel to ^<^; join e and d\ bisect ea in /; join / and c, 
intersecting e d in i\ bisect ic in o\ from o draw og at right 
angles to ic, meeting ^<^ extended to g\ join i and ^, cutting 
the transverse axis in r ; make Jij equal to kg, and Ji k equal 
to Jir\ from j, through r and /t, draw/ 7;^ RvAjn; also, from 
g, through k, draw gl; upon g and/, with gc for radius, 
describe the arcs i/ and ;;/;/; upon r and /c, with ra for 




Fig. 412. 

radius, describe the arcs ini^^nd ln\ this will complete the 
figure. 

When the axes are proportioned to one another, as at 2 
to 3, the extremities, c and d^ of the shortest axis, will be 
the centres for describing the arcs il and ni n ; and the inter- 
section oi e d with the transverse axis will be the centre for 
describing the arc in, z, etc. x\s the elliptic curve is contin- 
ually changing its course from that of a circle, a true ellipsis 
cannot be described with a pair of compasses. The above, 
therefore, is only an approximation. 

564. — To a^raw an Oval in the Proportion Seven by 
Bfine. — Let cd {Fig. 412) be the given conjugate axis. Bisect 



592 



PRACTICAL GEOMETRY. 



c d in 0, and through o draw ab "dX right angles \.o c d\ bisect 
CO in c ; upon o, with oc for radius, describe the circle efgh ; 
from e, through Ji and /, draw cj and ci\ also, from ^, 
through Ji and /", draw ^/^ and gl\ upon ^, with gc for 
radius, describe the arc kl\ upon r, with ed for radius, de- 
scribe the arc j i ; upon h and /, with ///^ for radius, describe 
the arcs j k and //; this will complete the figure. 




Fig. 413. 

This is an approximation to an elHpsis ; and perhaps no 
method can be found by which a well-shaped oval can be 
drawn with greater facility. By a little variation in the 
process, ovals of different proportions may be obtained. If 
quarter of the transverse axis is taken for the radius of the 
circle efgh, one will be drawn in the proportion five by 
seven. 




Fig. 414. 



555. — To Draiv a Tangent to an Ellipsis. — Let abed 
{Fig. 413) be the given ellipsis, and d the point of contact. 
Find the foci {Art. 548) / and /, and from them, through d,f^ 
draw f e and fd\ bisect the angle {Art. 506) edo with the 
line sr\ then sr will be the tangent required. 



TO FIND THE AXES OF AN ELLIPSE. 



593 



656. — An ElSipsis witti u Tang^ent Oivcn, to Detect the 
Point of Contact. — Let ci gb f {Fig. 414) be the given ellip- 
sis and tangent. Through the centre c draw a b parallel to 
the tangent; anywhere between e and / draw cd parallel to 
tib\ bisect cd in o\ through and e draw fg\ then g will 
be, the point of contact required. 

657. — A Diameter of an Ellipsis Given, to Find its 

Conjugate. — Let ab (Fig. 414) be the .given diameter. Find 
the line fg by the last problem ; then f g will be the diam- 
eter required. 

568. — Any Diameter and its Conjugate being Given, to 
Ascertain tlie Two Axes, and thence to Describe the ellipsis. 

— Let a b and c d {Fig. 415) be the given diameters, conjugate 




Fig. 415. 



to one another. Through c draw c f parallel \.o a b -, from v 
draw eg at right angles to cf\ make r^ equal to aJi or lib\. 
join g and Ji\ upon g^ with gc for radius, describe the arc 
ikcj\ upon Ji, with the same radius, describe the arc In-^ 
through the intersections / and ;/ draw no., cutting the tan- 
gent e f in ; upon <?, with ^^for radius, describe the semi- 
dipcle eigf\ join e and gy also ^and /", cutting the arc icj 
in k and / ; from e^ through Ji, draw e m, also from /, through 
h, draw fp; from k and / draw kr and ts parallel to gh.. 



594 



PRACTICAL GEOMETRY 



cutting em in r, and // in s^\ make km equal to Jir, and hp 
equal \.o hs\ then rin and s p will be the axes required, by 
which the ellipsis may be drawn in the usual way. 

659, — To Describe an Ellipsis, whose Axes shall be 
i*roportionate to the Axes of a Larger or l^malEer Given 
One. — Let ac b d {Fig. 416) be the given ellipsis and axes, and 




Fig. 416. 

/y the transverse axis of a proposed smaller one. Join a and 
c\ from i draw ie parallel to <^^ ; make of equal to oc ; then 
ef will be the conjugate axis required, and will bear the 
same proportion to ij as c d doQs to a b. (See Art. 541.) 

560. — To Describe a Parabola by Intersection of Lines. 

— Let ml {Fig. 417) be the axis and height (see Fig. 404) and 

1 2 3 / .'J 2 1 




dd a double ordinate and base of the proposed parabola. 
Through / draw a a parallel to dd ; through d and d draw 
da and da parallel to ml\ divide ad and dm, each into a 
like number of equal parts ; from each point of division in 



TO DESCRIBE AN HYPERBOLA. 



595 



dm draw the lines i i, 22, etc., parallel to 111 1\ from each 
point of division in da draw lines to /; then a curve traced 
through the points of intersection <?, 0, and 0, will be that of 
a parabola. 

Another method. Let in I {Fig. 418) be the axis and height, 
and dd the base. Extend ml and make la equal to ;;//; 
join a and d, and a and d\ divide ad and ad, each into a 
like number of equal parts, as at i, 2, 3, etc. ; join i and i, 2 
and 2, etc., and the parabola will be completed. (See Arts. 
460 to 472.) 

561. — To Describe an Ifyperbola by Intersection of 
Lrines. — Let ro {Fig. 419) be the height,// the base, and nr 
the transverse axis. (See Fig. 404.) Through r draw a a 




Fig. 41! 




1 2 3 <; 3 3 1 / 
Fig. 419. 



parallel to // ; from / draw ap parallel to ro\ divide ap 
and pa, each into a like number of equal parts; from each 
of the points of division in the base, draw lines to n ; from 
each of the points of division in ap, draw lines to r; then 
a curve traced through the points of intersection 0, 0, etc., 
will be that of an hyperbola. 

The parabola and hyperbola afford handsome curves for 
various mouldings. (See Figs. 191 to 205 ; 222 to 224; 241 
and 242 ; also note to Art. 318.) 



SECTION XVII.— SHADOWS. 

562. — Tlie Art of DraAving consists in representing 
solids upon a plane surface, so that a curious and nice ad- 
justment of lines is made to present the same appearance to 
the eye as does the human figure, a tree, or a house. It is 
by the effects of light, in its reflection, shade, and shadow, 
that the presence of an object is made known to us; so 
upon paper it is necessary, in order that the delineation 
may appear real, to represent fully all the shades and shad- 
ows that would be seen upon the object itself. In this sec- 
tion I propose to illustrate, by a few plain examples, the 
simple elementary principles upon which shading, in archi- 
tectural subjects, is based. The necessary knowledge of 
drawing, preliminary to this subject, is treated of in Section 
XV., from Arts. 487 to 498. 

563. — The Inclination of Ihc ELinc of §liac1o^r. — This 
is always, in architectural drawing, 45 degrees, both on the 
elevation and on the plan ; and the sun is supposed to be 
behind the spectator, and over his left shoulder. This can 
be illustrated by reference to Fig. 420, in which A repre- 
sents a horizontal plane, and B and C two vertical planes 
placed at right angles to each other. A represents the plan, 
(7 the elevation, and B a vertical projection from the eleva- 
tion. In finding the shadow of the plane B, the line ^ ^ is 
drawn at an angle of 45 degrees with the horizon, and the 
line^^ at the same angle with the vertical planed. The 
plane B being a rectangle, this makes the true direction of 
the sun's rays to be in a course parallel to db, which direc- 
tion has been proved to be at an angle of 35 degrees and 
16 minutes with the horizon. It is convenient, in shading, 
to have a set-square with the two sides that contain the 



CONVENTIONAL PLANES OF SHADOW. 



597. 



right angle of equal length ; this will make the two acute 
angles each 45 degrees, and will give the requisite bevel 
when worked upon the edge of the T-square. One reason 
why this angle is chosen in preference to another is that 
when shadows are properly made upon the drawing by it, 
the depth of every recess is more readily known, since the 
breadth of shadow and the depth of the recess will be equal. 
To distinguish between the terms shade and shadow^ it will 
be understood that all such parts of a body as are not ex- 
posed to the direct action of the sun's rays are in shade ; 




Fic;. 420. 



while those parts which are deprived of light by the inter- 
position of other bodies are in shadozv. 



564.— To Find the L,ine of l^liadow on Mouldings and 
other Horizontally Straig[ht Projections. — Figs. 421, 422, 
423, and 424 represent various mouldings in elevation, re- 
turned at the left, in the usual manner of miterins: around a 
projection. A mere inspection of the figures is sufficient to 
see how the line of shadow is obtained, bearing in mind that 
the ray a b is drawn from the projections at an angle of 45 



598 



SHADOWS. 



degrees. When there is no return at the end, it is neces- 
sary to draw a section, at any place in the length of the 
mouldings, and find the line of shadow from that. 

565. — To Find tlie L.ine of Sliadow Cast toy a j^helf. — In 

Fig. 425, A is the plan and B is the elevation of a shelf 
attached to a wall. From a and c draw a b and c d, accord- 
ing to the angle previously directed ; from b erect a per- 
pendicular intersecting c d-^td-, from /t^ draw dc parallel to 




V^WIW fl llipHl pi WWl lll il ll l l Bmp i Vmill ffl pWWM^ 




Fig. 421. 



Fig. 422. 





Fig. 423. 



Fig. 424. 



the shelf; then the lines cd and de w411 define the shadow 
cast by the shelf. There is another method of finding the 
shadow, without the plan A. Extend the lower line of the 
shelf toy, and make <:/ equal to the projection of the shelf 
from the w^all ; from/ draw /^ at the customary angle, and 
from c drop the vertical line eg intersecting fg'^tg-, from 
g draw ge parallel to the shelf, and from c draw c d Tit the 
usual angle; then the lines cd and dc will determine the 
extent of the shadow as before. 



SHADOWS OF STRAIGHT AND OBLIQUE SHELVES. 599 

666.— To Find llic Shadow Cast by a Slielf ^vliich i« 
Wider at one End than at tlie Other. — In Fig. 426, A is the 
plan, and B the elevation. Find the point d, as in the pre- 




FiG. 425. 



vious example, and from any other point in the front of the 
shelf, as a^ erect the perpendicular <7;^; from a and e draw^Z' 
and e c, at the proper angle, and from b erect the perpendicu- 




Fig. 426. 



lar bc^ intersecting ^^ in c\ from d, through c, draw do\ 
then the hues id a,nd ^^ will give the limit of the shadow 
cast by the shelf. 



6oo 



SHADOWS. 



567.— To Find tlic Shadow of a Shelf having: one End 
Aeute or Obtuse Angled. — Fig. 427 shows the plan and ele- 
vation of an acute-angled shelf. Find the line eg as before ; 





Fig. 427. 

from a erect the perpendicular ab\ join b and e\ then be 
and ^^ will define the boundary of shadow. 

568.— To Find the Shadow Cast by an Inclined Shelf.— 

In Fig. 428 the plan and elevation of such a shelf are shown, 
having also one end wider than the other. Proceed as di- 




FiG 428. 

rected for finding the shadows of Fig. 426, and find the points 
d and c ; then ad and dc will be the shadow required. If 
the shelf had been parallel in width on the plan, then the 
line dc would have been parallel with the shelf a b. 



SHADOWS OF INCLINED AND CURVED SHELVES. 



60 1 



569.— To Find tlic ISSsado^v Cast by a Slaelf Inclined in 
its Vertical Section either Upward or Down^vard. — From 
a {Figs. 429 and 430) draw a b Vit the usual angle, and from b 
draw be parallel with the shelf; obtain the point c by draw- 




FiG. 429. 



Fig. 430. 



ing a line from d at the usual angle. In Fig. 429 join e and 
/; then ic and cc will define the shadow. In Fig: 430, from 
o draw oi parallel with the shelf ; join i and e ; then ie and 
fc will be the shadow required. 

The projections in these several examples are bounded 





Fig. 431. 



Fig. 432. 



by straight lines ; but the shadows of curved lines may be 
found in the same manner, by projecting shadows from sev- 
eral points in the curved line, and tracing the curve of 
shadow through these points. {Figs. 431 and 432.) 



6o2 



SHADOWS. 



570.— To Find USac Sharto\^' of a SBaclf liaviiag iis Front 
Edg^c, or End, Curved on llic PJan, — In Figs. 431 and 432 
A and ^4 show an example of each kind. From several 
points, as a, a, in the plan, and from the corresponding points 
0, o in the elevation, draw rays and perpendiculars intersect- 





V 




■1,1 


% 


" e 


Illlllllllllllllll'l.l 


Mmi!l!IIllIlI 


i|:i!in!i;iiii( 




/ 




/ 



Fig. 433. 

ing at r, c, etc. ; through these points of intersection trace 
the curve, and it will define the shadow. 

571.— To Find the Shadow of a Shelf Curved in the Efie- 
vation. — In Fig. 433 find the points of intersection, r, c and 





Fig. 434. 

e, as in the last examples, and a curve traced through them 
will define the shadow. 

The preceding examples show how to find shadows when 
cast upon a vertical plajie ; shadows thrown upon curved sur- 
faces are ascertained in a similar manner. {Fig- 434.) 



SHADOW UPON AN INCLINED WALL. 



603 



572.— To Find the Shadow Cast upon s& Cylindrical 
IVall toy a Projection of any Kind. — By an inspection of 
Fig. 434, it will be seen that the only difference between this 
and the last examples is that the rays in the plan die against 
the circle ab^ instead of a straight line. 



^^ ^fiiiiiiiiiii 



Fig. 435. 

573-— To Find the Shado^v Cast toy a Shelf upon an In- 
clined Wall. — Cast the ray ab {Fig. 435) from the end of the 
shelf to the face of the wall, and from b draw /; c parallel to 
the shelf; cast the r^iyde from the end of the shelf; then 
the lines de and e c will define the shadow. 




Fig. 436. 

These examples might be multiplied, but enough has 
been given to illustrate the general principle by which shad- 
ows in all instances are found. Let us attend now to the 
application of this principle to such familiar objects as are 
likely to occur in practice. 



634 



SHADOWS. 



S74-- — To Find tlie §liado^r of a Projecting Horizontal 
Beam. — From the points a, a, etc. {Fig. 436), cast rays upon 
the wall ; the intersections c, c, c of those rajs with the per- 
pendiculars drawn from the plan will define the shadow. If 
the beam be inclined, either on the plan or elevation, at any 
angle other than a right angle, the difTerence in the manner 




"""""""I 



Fig. 437. 



of proceeding can be seen by reference to the preceding 
examples of inclined shelves, etc. 

575. — To Find tlie glaadoAv in a Reees§. — From the point 
a {Fig. 437) in the plan, and b in the elevation, draw the rays 
ac'^wdi be; from c erect the perpendicular ce, and from e 




Fig. 438. 

draw the horizontal line cd ; then the lines cc and ed will 
show the extent of the shadow. This applies only where 
the back of the recess is parallel with the face of the wall. 

_ 676.— To Find tlie Sliadour in a Reec§s, ii^lien llie Face 
of the \l^aH is Inclined^ and the Back of the Recess is 
Vertical. — In Fig. 438, A shows the section and B the eleva- 



SHADOW IN A FIREPLACE. 



605 



tion of a recess of this kind. From b, and from any other 
point in the line b a, as a, draw the rays be and ae ; from c, 
a, and e draw the horizontal lines eg, af, and eh; from d 




Fio. 439. 

and /cast the rays di and fh ; from /, through //, draw is; 
then i- i and ig will define the shadow. 

677._To Find llie Shadow in a Fireplace.— From a and 
b {Fig, 439) cast the rays a e and b e, and from c erect the 




llllllll'l!ll(llJ,||[fmilinL|l!H!'TJ 



Fig. 440. 

perpendicular ^^'; from ^ draw the horizontal line eo, and 
join and ^; then cr, r^, and od will give the extent of the 
shadow. 



6o6 



SHADOWS. 



578.— To Find tfiae l^liado^v of a IVIouBdcd Window-L.iii- 

tcl. — Cast rays from the projections a, o, etc., in the plan 
{Fig. 440), and d^ e, etc., in the elevation, and draw the usual 
perpendiculars intersecting the rays at i, 2, and i\ these in- 
tersections connected, and horizontal lines drawn from them, 
will define the shadow. The shadow on the face of the lin- 
tel is found by casting a ray back from i to Sy and drawing 
the horizontal line s n. 

579. — To Find the IShadow Cast by tlie Mosing of a Step. 

■ — From a {Fig. 441) and its corresponding point c, cast the 



ifiiiiiiiiiiiiiiiiiiiiiiiniiiiiiniii ii' 


iiiiitniininiiiM iiiiiiil 








% 


// 




\ll 


y 



I 



Fig. 441. 

rays a b and cd, and from b erect the perpendicular <^ ^^ ; tan- 
gical to the curve at c cast the ray e f, and from c drop the 
perpendicular e 0, meeting the mitre-line a g va o\ cast a ray 
from o to /, and from /erect the perpendicular if\ from // 
draw the ray Ji k ; from / to d and from d to k trace the 
curve as shown in the figure ; from k and //- draw the hori- 
zontal lines kn and hs\ then the limit of the shadow will be 
completed. 



580.— To Find the Shadow Thrown by a Pedestal upon 
Steps. — From a {Fig. 442) in the plan, and from c in the ele- 
vation, draw the rays ab and ce ; then ao Avill show the ex- 



SHADOWS ON STEPS AND COLUxMNS. 



607 



tent of the shadow on the first riser, as at A ; /,^will deter- 
mine the shadow on the second riser, as at /J; cd gives the 
amount of shadow on the first tread, as at C, and h i that on 
the second tread, as at Z^ ; which completes the shadow of 




Fig. 442. 

the left-hand pedestal, both on the plan and elevation. A 
mere inspection of the figure will be sufficient to show how 
the shadow of the right-hand pedestal is obtained. 





Fig. 443. 



Fig. 444. 



581.— To Find the Shadow Tlirowii on a Column by a 
$4quare Atoacu§. — From a and b {Fig. 443) draw the rays ac 
and ^^, and from ^ erect the perpendicular ^ r ; tangical to 
the curve at d draw the ray df, and from Ji, corresponding 
to /in the plan, draw the ray Jio', take any point between a 
and/, as /, and from this, as also from a corresponding point 



6o8 



SHADOWS. 



71, draw the rays ir and ns\ from r and from d erect the 
perpendiculars rs and ^^ ; through the pomts e, s, and o 
trace the curve as shown in the figure ; then the extent of 
the shadow will be defined. 




( (.(((((( c -^=^y^ 




Fig. 445. 



682- — To Find the Shadow Thrown on a Column toy a 
Circular Abacus. — This is so nearly like the last example 
that no explanation will be necessary, farther than a refer- 
ence to the preceding article. 



SHADOWS ON THE CAPITAL OF A COLUMN. 



609 



683-— To Find the ^Iaa<1ow§ on the Capital of a Column. 

— This may be done according to the principles explained 
in the examples already given ; a quicker way of doing it, 
however, is as follows : if we take into consideration one 
ray of light in connection with all those perpendicularly 
under and over it, it is evident that these several rays would 
form a vertical plane, standing at an angle of 45 degrees 
with the face of the elevation. Now we may suppose the 
column to be sliced, so to speak, with planes of this nature — 
cutting it in the lines a b, c d, etc. {Fig. 445), and, in the ele- 




FiG. 446. 



vation, find by squaring up from the plan, the lines of section 
which these planes would make thereupon. For instance : 
in finding upon the elevation the line of section a b, the plane 
cuts the ovolo at e, and therefore / will be the correspond- 
ing point upon the elevation ; // corresponds with g, i withy", 
with s, and / with b. Now, to find the shadows upon this 
line of section, cast from m the ray mn, from // the ray h 0, 
etc. ; then that part of the section indicated by the letters 
m / i jt, and that part also between /i and o will be under 



6io 



SHADOWS. 



shadow. By an inspection of the figure, it will be seen that 
the same process is applied to each line of section, and in 
that way the points /, r, t, 2i, v, zu, x, as also i, 2, 3, etc., are 




Fig. 447. 



successively found, and the lines of shadow traced through 
them. 

Fig. 446 is an example of the same capital w^ith all the 
shadows finished in accordance with the lines obtained on 
Mg. 445. 



SHADOW OF A COLUMN OX A WALL. 



6ll 



584o— 'S^w Fisicl I8ac Slaiidow Thrown oss si Vertical ^Valll 
by a i:o3siiBiia ami Eiilal>latur(' S^laiidiii^ bib Advance of sasd 

Wall. — Cast rays from a and /; (Z^?^. 447). and find the point 
c as in the previous examples ; from d draw the ray {^e, and 
from e the horizontal line e/; tangical to the curve at ^ and 
// draw the rays j^j and /i /, and from z and /erect the per- 
pendiculars // andyX'; from ;// and 71 draw the rays 7/1/ and 
;//', and trace the curve between /-and/; cast a ray from oto 
/, a vertical line from /to s, and through jr draw the horizon- 
tal line St ; the shadow as required will then be completed. 




Fig. 448. 

Fio-. 448 is an example of the same kind as the last, witli 
all the shadows filled in, according to the lines obtained in 
the preceding figure. 

5 85. — Shadows on a Cornice. — F/o^s. 449 and 450 arc 

examples of the Tuscan cornice. The manner of obtaining 
the shadows is evident. 



586.— Reflected V^ig;lit. — In shading, the finish and life of 
an object depend much on reflected light. This is seen to 
advantage in Fzg: 446, and on the column in Fig: 448. Re- 



6l2 



SHADOWS. 



fleeted rays are thrown in a direction exactly the reverse 
of direct rays ; therefore, on that part of an object which is 
subject to reflected light, the shadows are reversed. The 




FITTI^'WrMWTOlllWITliraTIW^ 




Fig. 449. 

fillet of the ovolo in Fig. 446 is an example of this. On the 
right hand side of the column, the face of the fillet is much 
darker than the cove directly under it. The reason of this 




is, the face of the fillet is deprived both of direct and re- 
flected light, whereas the cove is subject to the latter. Other 
instances of the effect of reflected light will be seen in the 
other examples. 



CONTENTS. 



PART I. 



SECTION I. — ARCH ITECTU RE 



Art. I. Building defined, p. 5. — 2. Antique Buildings ; Tower of Babel, 
p. 5. — 3. Ancient Cities and Monuments, p. 6. — 4. Architecture in Greece, 
p. 6. — 5, Architecture in Rome, p. 7. — 6. Rome and Greece, p. 8. — 7, Ar- 
chitecture debased, p. g. — 8, The Ostrogoths, p. 9. — 9. The Lombards, p. 10. 
— 10, The Byzantine Architects, p. 10. — 11. The Moors, p. 10. — 12, The 
Architecture of England, p. 11. — 13, Architecture Progressive, p. 12. — 14, 
Architecture in Italy, p. 12. — 15, The Renaissance, p. 13. — 16. Styles of Ar- 
chitecture, p. 13. — 17. Orders, p. 14. — 18. The Stylobate, p. 14. — 19, The 
Column, p. 14. — 20, The Entablature, p. 14. — 21, The Base, p. 14 — 22, 
The Shaft, p. 15.— 23. The Capital, p. 15.— 24. The Architrave, p. is.— 25. 
The Frieze, p. 15. — 26. The Cornice, p. 15. — 27. The Pediment, p. 15. — 28. 
The Tympanum, p. 15. — 29. The Attic, p. 15. — 30. Proportions in an Order, 
p. 15.— 31. Grecian Styles, p. 16.— 32. The Doric Order, p. 16.— 33. The 
Intercolumniation, p. 17. — 34, The Doric Order, p. 19 — 35. The Ionic 
Order, p. 19. — 36, The Intercolumniation, p. 20. — 37. To Describe the Ionic 
Volute, p. 20. — 38, The Corinthian Order, p. 23. — 39, Persians and Carya- 
tides, p. 24. — 40, Persians, p. 24. — 41, Caryatides, p. 26. — 42, Roman 
Styles, p. 26. — 43, Grecian Orders modified by the Romans, p. 27. — 44, The 
Tuscan Order, p. 30. — 45, Egyptian Style, p. 30. — 46. Building in General, 
p. 33. — 47. Expression, p. 35. — 48. Durability, p. 37. — 49. Dwelling- 
Houses, p. 37. — 50. Arranging the Stairs and Windows, p. 42. — 51. Prin- 
ciples of Architecture, p. 44. — 52. Arrangement, p. 44. — 53. Ventilation, p. 
45- — 54. Stability, p. 45. — 55. Decoration, p. 46.-56. Elementary Parts of 
a Building, p. 46. — 57, The Foundation, p. 47. — 58, The Column, or Pillar, 
p. 47.— 59, The Wall, p. 48.— 60, The Reticulated Walls, p. 49.— 61, The 
Lintel, or Beam, p. 49. — 62, The Arch, p. 50. — 63, Ilooke's Theory of an 
Arch, p. 50. — 64. Gothic Arches, p. 51.— 65, Arch : Definitions ; Principles, 
p. 52.-66. An Arcade, p. 52.-67, The Vault, p. 52.-68. The Dome, p. 53. 
—69, The Roof, p. 54. 



6 14 CONTEXTS. 

SECTION II.— CONSTRUCTION. 

Art. 70. Construction Essential, p. 56. — 71. Laws of Pressure, p. 57. — 
7*2, Parallelogram of Forces, p. 59. — 73, The Resolution of Forces, p. 59. — 
74. Inclination of Supports Unequal, p. Co. — 75. The Strains Exceed the 
Weights, p. 61. — 76. Minimum Thrust of Rafters, p. 62. — 77, Practical 
Method of Determining Strains, p. 62. — 7§. Horizontal Thrust, p. 63. — 79, 
Position of Supports, p. 65. — 80, The Composition of Forces, p. 66. — 81. 
Another Example, p. 67. — 82, Ties and Struts, p. 68. — 83, To Distinguish 
Ties from Struts, p. 69. — 84. Another Example, p. 70. — 85, Centre of Gravity, 
p. 71.— 86. Effect of the Weight of Inclined Beams, p. 72.-87, Effect of 
Load on Beam, p. 74. — 88. Effect on Bearings, p. 75. — 89. Weight-Strength, 
p, 76. — 90. Quality of Materials, p. 76. — 91. Manner of Resisting, p. 77. — 
92. Strength and Stiffness, p. 78. — 93. Experiments : Constants, p. 78. — 94, 
Resistance to Compression, p. 79. — 95. Resistance to Tension, p. 81. — 96. 
Resistance to Transverse Strains, p. 83. — 97, Resistance to Compression, p. 
85. — 98, Compression Transversely to the Fibres, p. 86. — 99. The Limit of 
Weight, p. 86.— 100, Area of Post, p. 86.-101, Rupture by Sliding, p. 87. 
—102, The Limit of Weight, p. 87.— 103. Area of Surface, p. 88.-104. 
Tenons and Splices, p. 88. — Il®5. Stout Posts, p. 89. — 106. The Limit of 
Weight, p. 89.— 107. Area of Post, p. 90. — 108. Area of Round Posts, p. 
90.— 109. Slender Posts, p. 91.— 110. The Limit of Weight, p. 91.— 111. 
Diameter of the Post: when Round, p. 92. — 112. Side of Post: when 
Square, p. 93. — 113. Thickness of a Rectangular Post, p. 95. — 114. Breadth 
of a Rectangular Post, p. 95. — 115. Resistance to Tension, p. 96. — 116. 
The Limit of Weight, p. 96. — 117. Sectional Area, p. 97. — 118. Weight of 
the Suspending Piece Included, p. 98. — 119. Area of Suspending Piece, 
p. 99. 

RESISTANCE TO TRANSVERSE STRAINS, 

Art. 120. Transverse Strains: Rupture, p. 99. — 121. Location of Mor- 
tises, p. 100. — 122. Transverse Strains : Relation of Weight to Dimensions, 
p. loi. — 123. Safe Weight : Load at Middle, p. 103.— 124. Breadth of Beam 
with Safe Load, p. 104. — 125. Depth of Beam with Safe Load, p. 104. — 126. 
Safe Load at any Point, p. 105. — 127. Breadth or Depth : Load at any Point, 
p. 106. — 128. Weight Uniformly Distributed, p. 107. — 129. Breadth or 
Depth: Load Uniformly Distributed, p. 108. — 130, Load per Foot Super- 
ficial, p. 109. — 131. Levers: Load at one End, p. no. — 132, Levers : Breadth 
or Depth, p. in. — 133, Deflection: Relation to Weight, p. 112. — 134, De- 
flection: Relation to Dimensions, p. 112. — 135, Deflection : Weight when at 
Middle, p. 114.— 136, Deflection : Breadth or Depth, Weight at Middle, p. 
114. — 137, Deflection : When Weight is at Middle, p. 116. — 138. Deflection : 
Load Uniformly Distributed, p. 116. — 139, Deflection : Weight when Uni- 
formly Distributed, p. 117.— -140, Deflection : Breadth or Depth, Load Uni- 
formly Distributed, p. 117. — 141, Deflection: When Weight is Uniformly 
Distributed, p. 118. — 142. Deflection of Lever, p. 119. — 143. Deflection of 
a Lever : Load at End, p. 120. — 144. Deflection of a Lever : Weight when at 
End, p. 120. — 145. Deflection of a Lever: Breadth or Depth, Load at End, 



CONTENTS. 615 

p. 121. — 146. Deflection of Levers : Weight Uniformly Distributed, p. 121. — 
147. Deflection of Levers with Uniformly Distributed Load, p. 122. — 148, 
Deflection of Levers : Weight when Uniformly Distributed, p. 122. — 149. 
Deflection of Levers : Breadth or Depth, Load Uniformly Distributed, p. 122. 

CONSTRUCTION IN GENERAL. 

Art. 150. Construction: Object Clearly Defined, p. 123. — 151. Floors 
Described, p. 124. — 152. Floor-Beams, p. 125. — 153. Floor-Beams for Dwell- 
ings, p. 127. — 154. Floor-Beams for First-Class Stores, p. 128. — 155. Floor- 
Beams : Distance from Centres, p. 129. — 156. Framed Openings for Chimneys 
and Stairs, p. 130. — 157. Breadth of Headers, p. 130. — 15§. Breadth of 
Carriage-Beams, p, 132. — 159. Breadth of Carriage-Beams Carrying Two 
Sets of Tail-Beams, p. 134. — 160. Breadth of Carriage-Beam with Well-Holo 
at Middle, p. 136. — 161. Cross-Bridging, or Herring-Bone Bridging, p. 137, 
— 162. Bridging: Value to Resist Concentrated Loads, p. 137. — 163. Gird- 
ers, p. 140, — 164. Girders : Dimensions, p. 141. 

FIRE-PROOF TIMBER FLOORS. 

Art. 165. Solid Timber Floors, p. 143.— 166. Solid Timber Floors for 
Dwellings and Assembly-Rooms, p. 143. — 167. Solid Timber Floors for First- 
Class Stores, p. 144. — 168. Rolled-Iron Beams, p. 145. — 169. Rolled-Iron 
Beams: Dimensions; Weight at Middle, p. 146. — 170. Rolled-Iron Beams: 
Deflection when Weight is at Middle, p. 147. — 171. Rolled-Iron Beams: 
Weight when at Middle, p. 148. — 172. Rolled-Iron Beams: Weight at any 
Point, p. 148. — 1173. Rolled-Iron Beams : Dimensions; Weight at any Point, 
p. 149. — fl74. Rolled-Iron Beams : Dimensions ; Weight Uniformly Distrib- 
uted, p. 149. — 175. Rolled-Iron Beams : Deflection ; Weight Uniformly Dis- 
tributed, p. 150. — 176. Rolled-Iron Beams : Weight when Uniformly Distrib- 
uted, p. 151. — 177. Rolled-Iron Beams: Floors of Dwellings or Assemblv- 
Rooms, p. 151. — 178. Rolled-Iron Beams: Floors of First-Class Stores, p. 
152. — 179. Floor-Arches: General Considerations, p. 153. — 180. Floor- 
Arches: Tic-Rods; Dwellings, p. 153. — 181, Floor-Arches: Tie-Rods; 
First-Class Stores, p. 153. 

TUBULAR IRON GIRDERS. 

Art. 182. Tubular Iron Girders: Description, p. 154. — 183. Tubular 
Iron Girders : Area of Flanges ; Load at Middle, p. 154. — 184. Tubular Iron 
Girders : Area of Flanges ; Load at an}- Point, p. 155. — 185. Tubular Iron 
Girders : Area of Flanges ; Load Uniformly Distributed, p. 156. — 186. Tu- 
bular Iron Girders: Shearing Strain, p. 157. — 187. Tubular Iron Girders: 
Thickness of Web, p. 158. — 188. Tubular Iron Girders for Floors of Dwell- 
ings, Assembly-Rooms, and Office Buildings, p. 159. — 189. Tubular Iron 
Girders for Floors of First-Class Stores, p. 160. 

CAST-IRON GIRDERS. 

Art. 190. Cast-Iron Girders: Inferior, p. 161. — 191. Cast-Iron Girder: 
Load at Middle, p. 161. — 192. Cast-Iron Girder: Load Uniformly Distributed, 



6l6 CONTENTS. 

p. 163.— 193. Cast-iron Bowstring Girder, p. 163.— 194. Substitute for the 
Bowstring Girder, p. 163. 

FRAMED GIRDERS. 

Art. 195. Graphic Representation of Strains, p. 165. — 196. Framed 
Girders, p. 166.— 197. Framed Girder and Diagram of Forces, p. 167.— 198. 
Framed Girders : Load on Both Chords, p. 171. — 199. Framed Girders: Di- 
mensions of Parts, p. 173. 

PARTITIONS. 

Art. 200. Partitions, p. 174. — 201. Examples of Partitions, p. 175. 

ROOFS. 

Art. 202. Roofs, p. 178. — 203. Comparison of Roof-Trusses, p. 178.— 
204. Force Diagram : Load upon Each Support, p. 179. — 205. Force Dia- 
gram for Truss in Fig. 59, p. 179. — 206. Force Diagram for Truss in Fig. 60, 
p. 180. — 207, Force Diagram for Truss in Fig. 61, p. 181. — 208. Force Dia- 
gram for Truss in Fig. 63, p. 183. — 209. Force Diagram for Truss in Fig. 64, 
p. 184. — 210. Force Diagram for Truss in Fig, 65, p. 1S5. — 211. Force Dia- 
gram for Truss in Fig. 66, p. 186. — 212. Roof-Truss : EfTect of Elevating the 
Tie-Beam, p. 187. — 213. Planning a Roof, p. 188. — 214. Load upon Roof- 
Truss, p. 189. — 215. Load on Roof per Superficial Foot, p. i8g. — 216. Load 
upon Tie-Beam, p. igo. — 217, Roof Weights in Detail, p. 191. — 218. Load 
per Foot Horizontal, p. 192. — 219. Weight of Truss, p. 192. — 220. Weight 
of Snow on Roofs, p. 193. — 221. Effect of Wind on Roofs, p. 193.— 222. 
Total Load per Foot Horizontal, p. 197. — 223. Strains in Roof Timbers 
Computed, p. 198. — 224. Strains in Roof Timbers Shown Geometrically, p. 
199. — 225. Application of the Geometrical System of Strains, p. 202. — 226. 
Roof Timbers : the Tie-Beam, p. 204.— 227. The Rafter, p. 205.— 228. The 
Braces, p. 208. — 229, The Suspension Rod, p. 210. — 230. Roof-Beams, 
Jack-Rafters, and Purlins, p. 211. — 231. Five Examples of Roofs, p. 212. — 
232. Roof-Truss with Elevated Tie-Beam, p. 2X4.-233. Hip-Roofs : Lines 
and Bevels, p. 215. — 234. The Backing of the Hip-Rafter, p. 216. 

DOMES. 

Art, 235. Domes, p. 216.— 236, Ribbed Dome, p. 217.— 237. Domes: 
Curve of Equilibrium, p. 218. — 238. Cubic Parabola Computed, p. 219. — 
239. Small Domes over Stairways, p. 220. — 240. Covering for a Spherical 
Dome, p. 221. — 241. Polygonal Dome : Form of Angle-Rib, p. 223. 



Art. 242, Bridges, p. 223. — 243. Bridges : Built-Rib, p. 224. — 244. 
Bridges : Fram.ed Rib, p. 226. — 245. Bridges : Roadway, p. 227, — 246. 
Bridges : Abutments, p. 227. — 247. Centres for Stone Bridges, p. 229. — 248. 
Arch Stones : Joints, p. 223. 

JOINTS. 

Art. 249. Timber Joints, p. 234. 



CONTENTS. 617 

SECTION III.— STAIRS. 

Art. 350. Stairs: General Requirements, p. 240. — 251. The Grade of 
Stairs, p. 241. — 252. Pitch-Board : Relation of Rise to Tread, p. 242. — 253. 
Dimensions of the Pitch-Board, p. 247. — 254. The String of a Stairs, p. 247. 
— 255. Step and Riser Connection, p. 248. 

PLATFORM STAIRS 

Art. 256. Platform Stairs: the Cylinder, p. 248.— 257. Form of Lower 
Edge of Cylinder, p. 249. — 25§. Position of the Balusters, p. 250. — 259. Wind- 
ing Stairs, p. 251. — 260. Regular Winding Stairs, p. 251.— 261. Winding 
Stairs : Shape and Position of Timbers, p. 252. — 262. Winding Stairs with 
Fl)^ers: Grade of Front String, p. 253. 

HAND-RAILING. 

Art. 26?l. Hand-Railing for Stairs, p. 256.-264. Hand-Railing: Defini- 
tions ; Planes and Solids, p. 257. — 265. Hand-Railing: Preliminar}- Consider- 
ations, p. 258. — 266. A Prism Cut by an Oblique Plane, p. 259.- 267. Form 
of Top of Prism, p. 259.-268. Face-Mould for Hand-Railing of Platform 
Stairs, p. 264. — 269. More Simple Method for Hand-Rail to Platform Stairs, 
p. 267.— 270. Hand-Railing for a Larger Cylinder, p. 271.— 271. Face- 
Mould without Canting the Plank, p. 272. — 272. Railing for Platform Stairs 
where the Rake meets the Level, p. 272. — 273. Application of Face-Moulds 
to Plank, p. 273. — 274. Face-Moulds for Moulded Rails upon Platform 
Stairs, p. 274. — 275. Application of Face-Moulds to Plank, p. 275.-276. 
Hand-Railing for Circular Stairs, p. 278. — 277. Face-Moulds for Circular 
Stairs, p. 282.-278. Face-Moulds, for Circular Stairs, p. 285.-279. Face- 
Moulds for Circular Stairs, again, p. 287. — 280. Hand-Railing for Winding 
Stairs, p. 289.— 281. Face-Moulds for Winding Stairs, p. 290.— 282. Face- 
Moulds for Winding Stairs, again, p. 293.-283. Face-Moulds : Test of Accu- 
racy, p. 295.-284. Application of the Face-Mould, p. 297.-285. Face-Mould 
Curves are Elliptical, p. 301.— 286. Face-Moulds for Round Rails, p. 303.— 
287. Position of the Butt Joint, p. 303.-288. Scrolls for Hand-Rails : Gen- 
eral Rule for Size and Position of the Regulating Square, p. 308.— 289. Cen- 
tres in Regulating Square, p. 308.— 290. Scroll for Hand-Rail Over Curtail 
Step, p. 309.— 291. Scroll for Curtail Step, p. 310.— 292. Position of Balus- 
ters Under Scroll, p. 310.— 293, Falling-Mould for Raking Part of Scroll, p. 
310.— 294. Face-Mould for the Scroll, p. 311.— 295. Form of Newel-Cap 
from a Section of the Rail, p. 312. — 296. Boring for Balusters in a Round Rail 
before it is Rounded, p. 313. 

SPLAYED WORK. 

Art. 297. The Bevels in Splayed Work, p. 314. 

SECTION IV.-DOORS AND WINDOWS. 

DOORS. 

Art. 298. General Requirements, p. 315. — 299. The Proportion between 
Width and Height, p. 315.-300. Panels, p. 316.— 301. Trimmings, p. 317. 
— 302. Hanging Doors, p. 317. 



mm 



6l8 CONTENTS. 



WINDOWS. 

Art. 303. Requirements for Light, p. 317. — 304. Window Frames, p. 
318. — 305. Inside Shutters, p. 319. — 306. Proportion: Width and Height, 
p. 319.— 307. Circular Heads, p. 320.— 308, Form of Soffit for Circular Win- 
dow Heads, p. 321. 

SECTION v.— MOULDINGS AND CORNICES. 

MOULDINGS. 

Art. 309. Mouldings, p. 323. -^-310. Characteristics of M®uldings, p. 
324. — 311. A Profile, p. 326. — 312. The Grecian Torus and Scotia, p. 326, — 
313. The Grecian Echinus, p. 327, — 314, The Grecian Cavetto, p. 327. — 
3S5. The Grecian Cyma-Recta, p. 327. — 316. The Grecian Cyma-Reversa, 
p. 328.— 317. Roman Mouldings, p. 329. — 31§. Modern Mouldings, p. 331. 

CORNICES. 

Art. 319, Designs for Cornices, p. 335. — 320. Eave Cornices Propor- 
tioned to Height of Building, p. 335. — 321. Cornice Proportioned to a given 
Cornice, p. 342. — 322. Angle Bracket in a Built Cornice, p. 343. — 323. Rak- 
ing Mouldings Matched with Level Returns, p. 344. 



PART II. 



SECTION VI.— GEOMETRY. 

Art. 324. Mathematics Essential, p. 347.-325. Elementary Geometry, 
p. 347. — 326. Definition— Right Angles, p. 348. — 327. Definition— Degrees 
in a Circle, p. 348. — 328. Definition — Measure of an Angle, p. 348. — 329. 
Corollary — Degrees in a Right Angle, p. 348.— 330. Definition — Equal 
Angles, p. 349. — 331. Axiom — Equal Angles, p. 349. — 332. Definition — 
Obtuse and Acute Angles, p. 349. — 333. Axiom — Right Angles, p. 349. — 
334. Corollary— Two Right Angles, p. 349.- 335, Corollary — Four Right 
Angles, p. 349. — 336. Proposition — Equal Angles, p. 350.-337. Propo- 
sition — Equal Triangles, p. 350. — 338. Proposition — Angles in Isosceles 
Triangle, p. 351. — 339. Proposition — Diagonal of Parallelogram, p. 351- — 
340. Proposition — Equal Parallelograms, p. 352. — 341. Proposition — Paral- 
lelograms Standing on the Same Base, p. 352.-342. Corollary — Parallelo- 
gram and Triangle, p. 353. — 343. Proposition — Triangle Equal to Quadrangle, 
P- 353.-344. Proposition— Opposite Angles Equal, p. 354-— 345. Proposi- 
tion—Three Angles of Triangle Equal to Two Right Angles, p. 354.-346. 
Corollary— Right Angle in Triangle, p. 354.-347. Corollary— Half a Right 



CONTENTS. 619 

Angle, p. 355.-348. Corollary — Right Angle in a Triangle, p. 355. — 349. 
Corollary — Two Angles Equal to Right Angle, p. 355. — 350, Corollary — Two 
Thirdsof a Right Angle, p. 355. — 351, Corollary — Equilateral Triangle, p. 355. 
— 352, Proposition — Right Angle in Semi-circle, p. 355. — 353, Proposition — 
The Square of the Hypothenuse Equal to the Squares of the Sides, p, 355. — 
354. Proposition — Equilateral Octagon, p. 357. — 355, Proposition — Angle 
at the Circumference of a Circle, p. 35S. — 356, Proposition — Equal Chords 
give Equal Angles, p. 35S. — 357, Corollary of Equal Chords, p. 359.-358. 
Proposition — Angle Formed by a Chord .and Tangent, p. 359. — 359, Propo- 
sition — Areas of Parallelograms, p. 360. — 360, Proposition — Triangles ol 
Equal Altitude, p. 361. — 361, Proposition— Homologous Triangles, p, 362. — 
362, Proposition — Parallelograms of Chords, p. 363. — 363, Proposition- 
Sides of Quadrangle, p. 364. 

SECTION VII.— RATIO, OR PROPORTION. 

Art. 364, Merchandise, p. 366.-365, The Rule of Three, p. 366.— 
366. Couples: Antecedent, Consequent, p. 367. — 367. Equal Couples : an 
Equation, p. 367. — 368. Equality of Ratios, p. 367. — 369. Equals Multiplied 
by Equals Give Equals, p. 367. — 370, Multiplying an Equation, p. 368. — 371, 
Multiplying and Dividing one Member of an Equation : Cancelling, p. 368. — 
372, Transferring a Factor, p. 369. — 373, Equality of Product: Means and 
Extremes, p. 369. — 374, Homologous Triangles Proportionate, p. 370. — 
375, The Steelyard, p. 371. —376, The Lever Exemplified by the Steelyard, 
p. 372. — 377. The Lever Principle Demonstrated, p. 375. — 378. Any One ot 
Four Proportionals may be Found, p. 377. 

SECTION VIIL— FRACTIONS. 

Art. 379. A Fraction Defined, p. 378. — 380. Graphical Representation 
of Fractions : Effect of Multiplication, p. 378. — 381. Form of Fraction 
Changed by Division, p. 3S0. — 382. Improper Fractions, p. 380. — 383. Re- 
duction of Mixed Numbers to Fractions, p. 381. — 384. Division Indicated by 
the Factors put as a Fraction, p. 381. — 385. Addition of Fractions having Like 
Denominators, p. 382. — 386. Subtraction of Fractions of Like Denominators, 
p. 383. — 387. Dissimilar Denominators Equalized, p. 383. — 388. Reduction 
of Fractions to their Lowest Terms, p. 384. — 389, Least Common Denomina- 
tor, p. 384. — 390. Least Common Denominator Again, p. 385. — 391, Frac- 
tions Multiplied Graphically, p. 3S6, — 392, Fractions Multiplied Graphically 
Again, p. 387. — 393. Rule for Multiplication of Fractions, and Example, p. 
387. — 394. Fractions Divided Graphically, p. 3S8, — 395. Rule for Division 
of Fractions, p. 389. 

SECTION IX.— ALGEBRA. 

Art. 396. Algebra Defined, p. 392. — 397. Example: Application, p. 
393.-398. Algebra Useful in Constructing Rules, p. 394.-399. Algebraic 
Rules are General, p. 394. — 400. Symbols Chosen at Pleasure, p. 395. — 401, 
Arithmetical Processes Indicated by Signs, p. 396. — 402, Examples in Addi- 



620 CONTENTS. 

tion and Subtraction : Cancelling, p. 398.— 403. Transferring a Symbol to the 
Opposite Member, p. 399.— 404. Signs of Symbols to be Changed when they 
j|re to be Subtracted, p. 400. — 405. Algebraic Fractions, Added and Sub- 
tracted, p. 403. — 406. The Least Common Denominator, p. 404. — 407. Alge- 
braic Fractions Subtracted, p. 405. — 408. Graphical Representation of Multi- 
plication, p. 408,— 409. Graphical Multiplication : Three Factors, p. 408. — 
410. Graphic Representation: Two and Three Factors, p. 409. — 411. Graph- 
ical Multiplication of a Binomial, p. 409.— 412. Graphical Squaring of a 
Binomial, p. 410. — 413. Graphical Squaring of the Difference of Two Fac- 
tors, p. 412. — 414. Graphical Product of the Sum and Difference of Two 
Quantities, p. 413. — 415. Plus and Minus Signs in Multiplication, p. 415. — 
416. Equality of Squares on Hypothenuse and Sides of Right-Angled Tri- 
angle, p. 416. — 417. Division the Reverse of Multiplication, p. 418. — 41§. 
Division: Statement of Quotient, p. 419. — 419. Division: Reduction, p. 419. 
—420. Proportionals : Analysis, p. 421. — 421, Raising a Quantity to any 
Power, p. 423. — 422. Quantities with Negative Exponents, p. 423. — 423. 
Addition and Subtraction of Exponential Quantities, p. 424. — 424. Multipli- 
cation of Exponential Quantities, p. 424. — 425. Division of Exponential 
Quantities, p. 424. — 426. Extraction of Radicals, p. 425. — 427. Logarithms, 
p. 425. — 42§. Completing the Square of a Binomial, p. 429. 

PROGRESSION. 

Art. 429. Arithmetical Progression, p. 432.— 430. Geometrical Progres- 
sion, p. 435. 

SECTION X.— POLYGONS. 

Art. 431. Relation of Sum and Difference of Two Lines, p. 439. — 432. 
Perpendicular, in Triangle of Known Sides, p. 440. — 433. Trigon : Radius of 
Circumscribed and Inscribed Circles : Area, p. 443. — 434. Tetragon : Radius 
of Circumscribed and Inscribed Circles: Area, p. 446. — 435. Hexagon : Ra- 
dius ot Circumscribed and Inscribed Circles : Area, p. 447. — 436. Octagon : 
Radius of Circumscribed and Inscribed Circles : Area, p. 449. — 437. Dodec- 
agon : Radius of Circumscribed and Inscribed Circles: Area, p. 452. — 438, 
Hecadecagon : Radius of Circumscribed and Inscribed Circles : Area, p. 455. 
— 439. Pol3^gons : Radius of Circumscribed and Inscribed Circles : Area, p. 
460. — 440. Polygons: Their Angles, p. 462. — 441. Pentagon: Radius of the 
Circumscribed and Inscribed Circles ; Area, p. 463. — 442. Polygons: Table 
of Constant Multipliers, p. 465. 



SECTION XL— THE CIRCLE. 

Art. 443. Circles : Diameter and Perpendicular : Mean Proportional, p. 
468. — 444. Circle : Radius from Given Chord and Versed Sine, p. 469. — 
445. Circle: Segment from Ordinates, p. 470. — 446. Circle: Relation of 
Diameter to Circumference, p. 472. — 447. Circle : Length of an Arc, p. 475. 
• — 448. Circle: Area, p. 475. — 449. Circle: Area of a Sector, p. 476. — 450. 
Circle : Area of a Segment, p. 477. 



CONTENTS. 621 

SECTION XIL— THE ELLIPSE. 

Art. 451. Ellipse: Definitions, p. 481. — 452. Ellipse: Equations to the 
Curve, p. 482. — 453. Ellipse: Relation of Axis to Abscissas of Axes, p. 484. 
— 454. Ellipse : Relation of Parameter and Axes, p. 485.-455. Ellipse : 
Relation of Tangent to the Axes, p. 4S5. — 456. Ellipse: Relation of Tangent 
with the Foci, p. 487. — 457. Ellipse : Relation of Axes to Conjugate Diam- 
eters, p. 487. — 45§. Ellipse : Area, p. 488.-459. Ellipse ; Practical Sugges- 
tions, p. 489. 

SECTION XIII.— THE PARABOLA. 

Art. 460. Parabola : Definitions, p. 492. — 461. Parabola : Equation to 
the Curve, p. 493. — 462. Parabola: Tangent, p. 493. — 463. Parabola: Sub- 
tangent, p. 496. — 464. Parabola: Normal and Subnormal, p. 496. — 465. 
Parabola : Diameters, p. 497. — 466. Parabola : Elements, p. 499. — 467. 
Parabola : Described Mechanically, p. 500, — 46§. Parabola : Described from 
Points, p. 502. — 469. Parabola : Described from Arcs, p. 503. — 470. Para- 
bola : Described from Ordinates, p. 504. — 471. Parabola: Described from 
Diameters, p. 507. — 472, Parabola : Area, p. 509. 



SECTION XIV.— TRIGONOMETRY. 

Art. 473. Right-Angled Triangles : The Sides, p, 510. — 474. Right- 
Angled Triangles : Trigonometrical Tables, p. 512. — 475, Right-Angled 
Triangles: Trigonometrical Value of Sides, p. 516. — 476. Oblique-Angled 
Triangles: Sines and Sides, p. 519. — 477. Oblique-Angled Triangles : First 
Class, p. 520. — 47§. Oblique-Angled Triangles: Second Class, p. 522. — 
479. Oblique-Angled Triangles : Sum and Difference of Two Angles, p. 523. 
— 480. Oblique-Angled Triangles : Third Class, p. 526. — 481. Oblique- 
Angled Triangles : Fourth Class, p. 528. — 482. Trigonometrical Formulaj : 
Right-Angled Triangles, p. 530. — 483. Trigonometrical Formulae : First 
Class, Oblique, p. 531. — 484. Trigonometrical Formulae: Second Class, 
Oblique, p. 532. — 485. Trigonometrical Formulae : Third Class, Oblique, p^ 
534. — 486. Trigonometrical Formulae : Fourth Class, Oblique, p. 534. 



SECTION XV.— DRAWING 

Art. 487. General Remarks, p. 536. — 488. Articles Required, p. 536. — 
489. The Drawing-Board, p. 536. — 490. Drawing-Paper, p. 537. — 491. To 
Secure the Paper to the Board, p. 537. — 492, The T-Square, p. 539. — 493. 
The Set-Square, p. 539. — 494. The Rulers, p. 540.— 495. The Instruments, 
p. 540.— 496, The Scale of Equal Parts, p. 540.— 497. The Use of the Set- 
Square, p. 541. — 498, Directions for Drawing, p. 542. 



622 CONTENTS. 

SECTION XVI.— PRACTICAL GEOMETRY. 
Art. 499. Definitions of Various Terms, p. 544. 

PROBLEMS. 

RIGHT LINES AND ANGLES. 

Art. 500. To Bisect a Line, p. 549. — 501. To Erect a Perpendicular, p. 
550.— 502. To let Fall a Perpendicular, p. 551. —503. To Erect a Perpen- 
dicular at the End of a Line, p. 551. — 504. To let Fall a Perpendicular near 
the End of a Line, p. 553. — 505. To Make an Angle Equal to a Given Angle, 
P- 553.— 506. To Bisect an Angle, p. 554. — 507. To Trisect a Right Angle, 
P- 554- — 508. Through a Given Point to Draw a Line Parallel to a Given 
Line, p. 555. — 509. To Divide a Given Line into any Number of Equal Parts, 
P- 555. 

THE CIRCLE, 

Art. 510. To Find the Centre of a Circle, p. 556.— 511. At a Given 
Point in a Circle to Draw a Tangent thereto, p. 557. — 512, The Same, with- 
out making use of the Centre of the Circle, p. 557. — 513. A Circle and a 
Tangent Given, to Find the Point of Contact, p. 558. — 514. Through any 
Three Points not in a Straight Line to Draw a Circle, p. 559. — 515. Three 
Points not in a Straight Line being Given, to Find a Fourth that Shall, with 
the Three, Lie in the Circumference of a Circle, p. 559. — 516. To Describe 
a Segment of a Circle by a Set-Triangle, p. 560. — 517. To Find the Radius of 
an Arc of a Circle when the Chord and Versed Sine are Given, p. 561. — 518. 
To Find the Versed Sine of an Arc of a Circle when the Radius and Chord 
are Given, p. 561. — 519, To Describe the Segment of a Circle by Intersection 
of Lines, p. 562. — 520. Ordinates, p. 563. — 521. In a Given Angle to De- 
scribe a Tanged Curve, p. 565. — 522. To Describe a Circle within any Given 
Triangle, so that the Sides of the Triangle shall be Tangical, p. 566. — 523. 
About a Given Circle to Describe an Equilateral Triangle, p. 566. — 524. To 
Find a Right Line nearly Equal to the Circumference of a Circle, p. 566. 



POLYGONS, ETC. 

Art. 525. Upon a Given Line to Construct an Equilateral Triangle, p. 
568. — 526. To Describe an Equilateral Rectangle, or Square, p. 568. — 527. 
Within a Given Circle to Inscribe an Equilateral Triangle, Hexagon, or Dodec- 
agon, p. 569. — 528. Within a Square to Inscribe an Octagon, p. 570.— 529. 
To Find the Side of a Buttressed Octagon, p. 571. — 530. Within a Given 
Circle to Inscribe any Regular Polygon, p. 572. — 531. Upon a Given Line to 
Describe any Regular Pol3'gon,p. 573. — 532. To Construct a Triangle whose 
Sides shall be severally Equal to Three Given Lines, p. 575. — 533. To Con- 
struct a Figure Equal to a Given Right-lined Figure, p. 575. — 534. To Make 
a Parallelogram Equal to a Given Triangle, p. 576. — 535. A Parallelogram 
being Given, to Construct Another Equal to it, and Having a Side Equal to a 



CONTENTS. 623 

Given Line, p. 576. — 536. To Make a Square Equal to two or more Given 
Squares, p. 577. — 537. To Make a Circle Equal to two Given Circles, p. 5S0. 
— 538. To Construct a Square Equal to a Given Rectangle, p. 5S1. — 539. To 
Form a Square Equal to a Given Triangle, p. 582. — 540. Two Right Lines 
being Given, to Find a Third Proportional thereto, p. 582. — 541. Three Right 
Lines being Given, to Find a Fourth Proportional thereto, p. 583. — 542. A 
Line with Certain Divisions being Given, to Divide Another, Longer or 
Shorter, Given Line in the Same Proportion, p. 583. — 543. Between Two 
Given Right Lines to Find a Mean Proportional, p. 584. 



CONIC SECTIONS. 

Art. 544. Definitions, p. 5S4. — 545. To Find the Axes of the Ellipsis, 
p. 585.-546. To Find the Axis and Base of the Parabola, p. 585.-547. To 
Find the Height, Base, and Transverse Axis of an Hyperbola, p. 585. — 54S. 
The Axes being Given, to Find the Foci, and to Describe an Ellipsis with a 
String, p. 586. — 549, The Axes being Given, to Describe an Ellipsis with a 
Trammel, p. 5S6.— 550. To Describe an Ellipsis by Ordinates, p. 588. — 551. 
To Describe an Ellipsis by Intersection of Lines, p. 58S. — 552. To Describe 
an Ellipsis by Intersecting Arcs, p. 590. — 553. To Describe a Figure Nearly 
in the Shape of an Ellipsis by a Pair of Compasses, p. 591. — 554. To Draw 
an Oval in the Proportion Seven by Nine, p. 591. — 555. To Draw a Tangent 
to an Ellipsis, p. 592. — 556. An Ellipsis with a Tangent Given, to Detect the 
Point of Contact, p. 593. — 557, A Diameter of an Ellipsis Given, to Find its 
Conjugate, p. 593. — 558. Any Diameter and its Conjugate being Given, to 
Ascertain the Two Axes, and thence to Describe the Ellipsis, p. 593. — 559, 
To Describe an Ellipsis, whence Axes shall be Proportionate to the Axes of 
a Larger or Smaller Given One, p. 594. — 560, To Describe a Parabola by 
Intersection of Lines, p. 594. — 561, To Describe an Hyperbola by Intersec- 
tion of Lines, p. 595. 



SECTION XVII.— SHADOWS. 

Art. 562. The Art of Drawing, p. 596.-563. The Inclination of the Line 
of Shadow, p. 596. — 564. To Find the Line of Shadow on Mouldings and 
other Horizontally Straight Projections, p. 597. — 565. To Find the Line of 
Shadow Cast by a Shelf, p. 598.-566. To Find the Shadow Cast by a Shelf 
which is Wider at one End than at the Other, p. 599. — 567. To Find the 
Shadow of a Shelf having one End Acute or Obtuse Angled, p. 600. — 568. 
To Find the Shadow Cast by an Inclined Shelf,- p. 600.— 569. To Find the 
Shadow Cast by a Shelf inclined in its Vertical Section either Upward or 
Downward, p. 601.— 570. To Find the Shadow of a Shelf having its Front 
Edge or End Curved on the Plan, p. 602. — 571, To Find the Shadow of a 
Shelf Curved in the Elevation, p. 602.— 572. To Find the Shadow Cast upon 
a Cylindrical Wall by a Projection of any Kind, p. 603. — 573. To Find the 
Shadow Cast by a Shelf upon an Inclined Wall, p. 603. — 574. To Find the 
Shadow of a Projecting Horizontal Beam, p. 604. — 575, To Find the Shadow 



624 CONTENTS. 

in a Recess, p. 604. — 576. To Find the Shadow in a Recess, when the Face of 
the Wall is Inclined, and the Back of the Recess is Vertical, p. 604. — 577, 
To Find the Shadow in a Fireplace, p. 605. — 578. To Find the Shadow of a 
Moulded Window-Lintel, p. 606. — 579. To Find the Shadow Cast by the 
Nosing of a Step, p. 606. — 580. To Find the Shadow Thrown by a Pedestal 
upon Steps, p. 6c6. — 581. To Find the Shadow Thrown on a Column by a 
Square Abacus, p. 607. — 582. To Find the Shadow Thrown on a Column by 
a Circular Abacus, p. 60S. — 583. To Find the Shadows on the Capital of a 
Column, p. 609. — 584. To Find the Shadow Thrown on a Vertical Wall by a 
Column and Entablature Standing in Advance of said Wall, p. 611. — 585o 
Shadows on a Cornice, p. 611. — 586. Reflected Light, p. 6ii. 



AMERICAN HOUSE CARPENTER. 



APPENDIX 



CONTENTS. 



PAGE. 
627 



Glossary 

Table of Squares, Cubes, and Roots ^3 

Rules for the Reduction of Decimals 

Table of Circles ; 

Table showing the Capacity of Wells, Cisterns, etc 653 

Table of the Weights of Materials ^54 



647 
649 



GLOSSARY. 



Terms not found here can he found in the lists of definitions in othey parts of this book, or in 

common dictionaries. 



Abacus. — The uppermost member of a capital. 

Abattoir. — A slaughter-house. 

Abbey. — The residence of an abbot or abbess. 

Abutment. — That part of a pier from which the arch springs. 

Acanthus. — A plant called in English bear's-breech. Its leaves are employed 
for decorating the Corinthian and the Composite capitals. 

Acropolis. — The highest part of a city ; generail)^ the citadel, 

Acroteria. — The small pedestals placed on the extremities and apex of a 
pediment, originally intended as a base for sculpture. 

Aisle. — Passage to and from the pews of a church. In Gothic aicliitecture, 
the lean-to wings on the sides of the nave. 

Alcove. — Part of a chamber separated by an estradc, or partition of columns. 
Recess with seats, etc., in gardens. 

Altar. — A pedestal whereon sacrifice was cflfered. In modern churches; the 
area within the railing in front of the pulpit. 

Alto-relievo. — High relief; sculpture projectingfrom a surface so as to appear 
nearly isolated. 

Amphitheatre. — A double theatre, employed by the ancients for the exhibi- 
tion of gladiatorial fights and other shows. 

Ancones. — Trusses employed as an apparent support to a cornice upon the 
flanks of the architrave. 

Annulet. — A small square moulding used to separate others ; the fillets in 
the Doric capital under the ovolo, and those which separate the flutings of col- 
umns, are known by this term. 

Ant(r. — A pilaster attached to a wall. 

Apiary. — A place for keeping beehives. 

Arabesque. — A building after the Arabian style. 

Areostyle.—hx\ intercolumniation of from four to five diameters. 

Arcade. — A series of arches. 

Arch. — An arrangement of stones or other material in a curvilinear form, so 
as to perform the office of a lintel and carry superincumbent weights. 

Architrave.— '\\\^\ part of the entablature which rests upon the capital of a 
column, and is beneath the frieze. The casing and mouldings about a door or 
window. 

^;r///7'^//.— The ceiling of a vault ; the under surface of an arch. 

^;-^.^. —Superficial measurement. An open space, below the level of the 
ground, in front of basement windows. 



62 8 APPENDIX. ~ 

Arsenal. — A public establishment for the deposition of arms and warlike 
stores. 

Astragal. — A small moulding consisting of a half-round with a fillet on each 
side. 

Attic. — A low story erected over an order of architecture. A low additional 
story immediately under the roof of a building. 

Aviary. — A place for keeping and breeding birds. 

Balcony. — An open gallery projecting from the front of a building. 

Baluster. — A small pillar or pilaster supporting a rail. 

Balustrade. — A series of balusters connected by a rail. 

Barge-course. — That part of the covering which projects over the gable of a 
building. 

Base. — The lowest part of a wall, column, etc. 

Basement-sto7y. — That which is immediately under the principal story, and 
included within the foundation of the building. 

Basso-relievo. — Low relief ; sculptured figures projecting from a surface one 
half their thickness or less. See Alto-relievo. 

Battering. — See Talus. 

Battlement. — Indentations on the top of a wall or parapet. 

Bay-window. — A window projecting in two or more planes, and not formi- 
ing the segment of a circle. 

Bazaar. — A species of mart or exchange for the sale of various articles of 
merchandise. 

Bead. — A circular moulding. 

Bed-mouldings. — Those mouldings which are between the corona and the 
frieze. 

Belfry. — That part of the steeple in which the bells arc hung ; anciently 
called campanile. 

Belvedere. — An ornamental turret or observatory commanding a pleasant 
prospect. 

Bozu-window. — A window projecting in curved lines. 

Bresstimvier. — A beam or iron tie supporting a wall over a gateway or other 
opening. 

Brick-nogging. — The brickwork between studs of partitions. 

Buttress. — A projection from a wall to give additional' strength. 

Cable. — A cylindrical moulding placed in flutes at the lower j)art of the col- 
umn. 

Camber. — To give a convexity to the upper surface of a beam. 

Campanile. — A tower for the reception of bells, usually, in Italy, separated 
from the church. 

Canopy. — An ornamental covering over a seat of state. 

Cantalivers. — The ends of rafters under a projecting roof. Pieces of wood 
or stone supporting the eaves. 

Capital. — The uppermost part of a column included between the shaft and 
the architrave. 

Caravansera. — In the East, a large public building for the reception of trav- 
ellers by caravans in the desert. 






GLOSSARY. 629 

Cctrpentry.—{¥ \:om the Latin carpeiitum, carved wood.) That department 
of science and art which treats of the disposition, the construction, and the 
relative strength of timber. The first is called descriptive, the second con- 
structive, and the last mechanical carpentry. 

Caryatides. — Figures of women used instead cf columns to support an 
entablature. 

Casino. — A small country-house. 

Castellated. — Built with battlements and turrets in imitation of ancient 
castles. 

Castle. — A building fortified for military defence. A house with towers, 
usually encompassed with walls and moats, and having a donjon, or keep, in 
the centre. 

Catacombs. — Subterraneous places for burying the dead. 

Cathedral. — The principal church of a province or diocese, wherein the 
throne of the archbishop or bishop is placed. 

Cavetto. — A concave moulding comprising the quadrant of a circle. 

Cemetery. — An edifice or area where the dead are interred. 

Cenotaph. — A monument erected to the memory of a person buried in 
another place. 

Centring. — The temporary woodwork, or framing, whereon any vaulted 
work is constructed. 

Cesspool. — A well under a drain or pavement to receive the waste water and 
sediment. 

Chamfer. — The bevelled edge of anything originally right angled. 

Chancel. — That part of a Gothic church in which the altar is placed. 

Chajttry. — A little chapel in ancient churches, with an endowment for one 
or more priests to say mass for the relief of souls out of purgatory. 

Chapel. — A building for religious worship, erected separately from a Church, 
and served by a chaplain, 

Chaplet. — A moulding carved into beads, olives, etc. 

Cijicttire. — The. ring, listel, or fillet, at the top and bottom of a column, 
which divides the shaft of the column from its capital and base. 

Circus. — A straight, long, narrow building used by the Romans for the ex- 
hibition of public spectacles and chariot races. At the present day, a building 
enclosing an arena for the exhibition of feats of horsemanship. 

Clere-story. — The upper part of the nave of a church above the roofs of the 
aisles. 

Cloister. — The square space attached to a regular monastery or large church, 
having a peristyle or ambulatory around it, covered with a range of buildings. 

Coffer-dam. — A case of piling, water-tight, fixed in the bed of a river, for the 
purpose of excluding the water while any work, such as a wharf, wall, or the 
pier of a bridge, is carried up. 

Collar-beain. — A horizontal beam framed between two principal rafters above 
the tie-beam. 

Colonnade. — A range of columns. 

Columbarium. — A pigeon-house. 

Column. — A vertical cylindrical support under the entablature of an order. 

Common-rafters. — The same 3.s Jack-rafters, which see. 



630 APPENDIX. 

Conduit. — A long, narrow, walled passage underground, for secret com- 
munication between different apartments. A canal or pipe for the conveyance 
of water. 

Conservatory. — A building for preserving curious and rare exotic plants. 

Consoles. — The same as ancones, which see. 

Contour. — The external lines which bound and terminate a figure. 

Convent. — A building for the reception of a society of religious persons. 

Coping. — Stones laid on the top of a wall to defend it from the weather, 
■ Corbels. — Stones or timbers fixed in a wall to sustain the timbers of a lloor 
or roof. 

Cornice. — Any moulded projection which crowns or finishes the part to 
which it is affixed. 

Corona. — That part of a cornice which is between the crown-moulding and 
the bed-mouldings. 

Cornucopia. — The horn of plenty. 

Corridor. — An open gallery or communication to the dififerent apartments of 
a house. 

Cove. — A concave moulding. 

Cripple-rafters . — The short rafters which are spiked to the hip-rafter of a 
roof. 

Crockets. — In Gothic architecture, the ornaments placed along tho angles of 
pediments, pinnacles, etc. 

Crosettes. — The same as ancones, which see. 

Crypt. — The under or hidden part of a building. 

Culvert. — An arched channel of masonry or brickwork, built beneath the 
bed of a canal for the purpose of conducting water under it. Any arched 
channel for water underground. 

Cupola. — A small building on the top of a dome. 

Curtail-step. — A step with a spiral end, usually the first of the flight. 

Ctisps. — The pendants of a pointed arch. 

Cyina. — An ogee. There are two kinds ; the cyma-recta, having the upper 
part concave and the lower convex, and the cyma-reversa, with the upper part 
convex and the lower concave. 

Dado. — The die, or part between the base and cornice of a pedestal. 

Dairy. — An apartment or building for the preservation of milk, and the 
manufacture of it into butter, cheese, etc, 

Dead-shoar. — A piece of timber or stone stood vertically in brickwork, to 
support a superincumbent weight until the brickwork which is to carry it 
has set or become hard. 

Decastyle. — A building having ten columns in front. 

Dentils. — (From the Latin, denies, teeth.) Small rectangular blocks used in 
the bed-mouldings of some of the orders. 

Diastyle. — An intercolumniation of three, or, as some say, four diameters. 

Die. — That part of a pedestal included between the base and the cornice ; it 
is also called a dado. 

Dodecastyle. — A building having twelve columns in front. 

Donjon. — A massive tower within ancient castles, to which the garrison 
might letrcat in case of necessity. 



GLOSSARY. 631 

Docks. — A Scotch name given to wooden brick'. 

Dormer. — A window placed on the roof of a house, the frame being placed 
vejlically on the rafters. 

Dormitory. — A sleeping- room. 

Dovecote. — A building for keeping tame pigeons. A cclumbarium. 

Echinns. — The Grecian ovolo. 

Elevation. — A geometrical projection drawn on a plane at right angles to 
the horizon. 

Entablatiire. — That part of an order which is supported by the columns ; 
consisting of the architrave, frieze, and cornice. 

Eiisiyle. — An intercolumniaiion of two and a quarter diameters. 

Exchange. — A building in which merchants and brokers meet to transact 
business. 

Extrados. — The exterior curve of an arch, 

Ea^ade. — The principal front of any building. 

Eace-mould. — The pattern for marking the plank out of which hand-railing 
is to be cut for stairs, etc. 

Facia, or Fascia. — A fiat member, like a band or broad fillet. 

Falliv.g-motdd. — The mould applied to the convex, vertical surface of the 
rail-piece, in order to form the back and under surface of the rail, and finish 
the squaring. 

Festoon. — An ornament representing a wreath of flowers and leaves. 

Fillet. — A narrow flat band, llstel, or annulet, used for the separation of 
one moulding from another, and to give breadth and firmness to the edges of 
mouldings. 

Flutes. — Upright channels on the shafts of columns. 

Flyers. — Steps in a flight ot stairs that are parallel to each other. 

Forum. — In ancient architecture a public market ; also, a place where the 
common courts were held and law pleadings carried on. 

Foundry. — A building in which various metals are cast into moulds or 
shapes. 

Fjieze. — That part of an entablature included between the architrave and 
the cornice. 

Gable. — The vertical, triangular piece of wall at the end of a roaf, from the 
level of the eaves to the summit. 

Gaijt. — A recess made to receive a tenon or tusk. 

Gallery. — A common passage to several rooms in an upper story. A long 
room for the reception of pictures. A platform raised on columns, pilasters, 
or piers. 

Girder. — The principal beam in a floor, for supporting the binding and 
other joists, whereby the bearing or length is lessened. 

Glyph. — A vertical, sunken channel. From their number, those in the 
Doric order are called triglyphs. 

Granary. — A building for storing grain, especially that intended to be 
kept for a considerable time. 



632 APPENDIX. 

Groin. — The line formed by the intersection of two arches, which cross each 
other at any angle. 

GuttcE. — The small cylindrical pendent ornaments, otherwise called drops, 
used in the Doric order under the triglyphs, and also pendent from the mutuli 
of the cornice. 

Gymnasium. — Originally, a place measured out and covered with sand for 
the exercise of athletic games ; afterward, spacious buildings devoted to the 
mental as well as corporeal instruction of youth. 

Hall. — The first large apartment on entering a house. The public room of 
a corporate body. A manor-house. 

Ham.—K house or dwelling-place. A street or village : hence Nolting- 
ham, Bucking//^w, etc. Hamlet, the diminutive of ham, is a small street or 
village. 

Helix. — The small volute, or twist, under the abacus in the Corinthian 
capital. 

Hem. — The projecting spiral fillet of the Ionic capital. 

Hexastyle. — A building having six columns in front. 

Hip-rafter. — A piece of timber placed at the angle made by two adjacent 
inclined roofs. 

Homestall. — A mansion-house, or seat in the country. 

Hotel, or Hostel. — A large inn or place of public entertainment. A large 
house or palace. 

Hot-house. — A glass building used in gardening. 

Hovel. — An open shed. 

Hut. — A small cottage or hovel, generally constructed of earthy materials, 
as strong loamy clay, etc. 

Impost. — The capital of a pier or pilaster which supports an arch. 
Intaglio. — Sculpture in which the subject is hollowed out, so that the im- 
pression from it presents the appearance of a bas-relief. 
Intercolumniation. — The distance between two columns. 
Intrados. — The interior and lower curve of an arch. 

Jack-rafters. — Rafters that fill in between the principal rafters of a roof ; 
called also common-rafters. 

Jail. — A place of legal confinement. 

Jambs. — The vertical sides of an aperture. 

Joggle-piece. — A post to receive struts. 

Joists. — The timbers to which the boards of a floor or the laths of a ceiling 
are nailed. 

Keep. — The same as donjon, which see. 
Key-stone. — The highest central stone of an arch. 

Kiln. — A building for the accumulation and retention of heat, in order to 
dry or burn certain materials deposited within it. 
King-post. — The centre-post in a trussed roof. 
Knee. — A convex bend in the back of a hand-rail. See Ramp. 



GLOSSARY. 633 

Lactariion. — The same as dairy, which see. 

Lantern. — A cupola having windows in the sides for lighting an apartnnent 
beneath. 

Larmier. — The same as corona, which see. 

Lattice. — A reticulated window for the admission of air, rather than light, 
as in dairies and cellars. 

Lever-boards. — Blind-slats ; a set of boards so fastened that they may be 
turned at any angle to admit more or less light, or to lap upon each other so 
as to exclude all air or light through apertures. 

Lintel. — A piece of timber or stone placed horizontally over a door, win- 
dow, or other opening. 

Listel. — The same diS, Jillet, which see. 

Lobby. — An enclosed space, or passage, communicating with the principal 
room or rooms of a house. 

Lodge. — A small house near and subordinate to the mansion. A cottage 
placed at the gate of the road leading to a mansion. 

Loop. — A small narrow window. Loophole is a term applied to the vertical 
series of doors in a warehouse, through which goods are delivered by means 
of a crane. 

Lnffer-boarding. — The same as lever-boards, which see. 

Lnthern. — The same as dormer, which see. 

ATansoletun. — A sepulchral building — so called from a very celebrated one 
erected to the memory of Mausolus, king of Caria, by his wife Artemisia. 

Metopa. — The square space in the frieze between the triglyphs of the Doric 
order. 

Mezzanine. — A story of small height introduced between two of greater 
height. 

Minaret. — A slender, lofty turret having projecting balconies, common in 
Mohammedan countries. 

Mimler. — A church to which an ecclesiastical fraternity has been or is 
attached. 

Moat. — An excavated reservoir of water, surrounding a house, castle, or 
town. 

Modillion. — A projection under the corona of the richer orders, resembling 
a bracket. 

Module. — The semi-diameter of a column, used by the architect as a meas- 
ure by which to proportion the parts of an order. 

Alonastcry. — A building or buildings appropriated to the reception of 
monks. 

Monopteron. — A circular colonnade supporting a dome without an enclos- 
ing wall. 

Mosaic. — A mode of representing objects by the inlaying of small cubes of 
glass, stone, marble, shells, etc. 

Mosque. — A Mohammedan temple or place of worship. 

Miilliojis. — The upright posts or bars which divide the lights in a Gothic 
window. 

Mnniment-Jiottse. — A strong, fire-proof apartment for the keeping and pres- 
ervation of evidences, charters, seals, etc., called muniments. 



634 APPENDIX, 

Museum. — A repository of natural, scientific, and literary curiosities or of 
works of art. 

Afutule. — A projecting ornament of the Doric cornice supposed to repre- 
sent the ends of rafters. 

Nave. — The main body of a Gothic church. 
Neiuel. — A post at the starting or landing of a flight of stairs. 
Niche. — A cavity or hollow place in a wall for the reception of a statue, 
vase, etc. 

N'ogs. — Wooden bricks. 

Nosing. — The rounded and projecting edge of a step in stairs. 

Ntmnery. — A building or buildings appropriated for the reception of nuns. 

Obelisk. — A lofty pillar of a rectangular form. 

Octastyle. — A building with eight columns in front. 

Odeiun. — Among the Greeks, a species of theatre wherein the poets and 
musicians rehearsed their compositions previous to the public production of 
them. 

Ogee. — See cyma. 

Orangery. — A gallery or building in a garden or parterre fronting the 
south. 

Oriel-window. — A large bay or recessed window in a hall, chapel, or other 
apartment. 

Ovolo. — A convex projecting moulding whose profile is the quadrant of a 
circle. 

Pagoda. — A temple or place of worship in India. 

Palisade. — A fence of pales or stakes driven into the ground. 

Parapet. — A small wall of any material for protection on the sides of 
bridges, quays, or high buildings. 

Pavilion. — A turret or small building generally insulated and comprised 
under a single roof. 

Pedestal. — A square foundation used to elevate and sustain a column, 
statue, etc. 

Pediment. — The triangular crowning part of a portico or aperture which 
terminates vertically the sloping parts of the roof ; this, in Gothic architecture, 
is called di gable. 

Penitentiary. — A prison for the confinement of criminals whose crimes arc 
not of a very heinous nature. 

Piazza. — A square, open space surrounded by buildings. This term is 
often improperly used to denoic^ portico. 

Pier. — A rectangular pillar without any regular base or capital. The up- 
right, narrow portions of walls between doors and windows are known by this 
term. 

Pilaster. — A square pillar, sometimes insulated, but more commonly en- 
gaged in a wall, and projecting only a part of its thickness. 

Piles. — Large timbers driven into the ground to make a secure foundation 
in marshy places, or in the bed of a river. 



GLOSSARY. 635 

Pillat. — A column of irregular form, always disengaged, and always deviat- 
ing from the proportions of the orders ; whence the distinction between a 
pillar and a column. 

Pinnacle. — A small spire used to ornament Gothic buildings. 

Planceer. — The same as soffit, which see. 

Plinth. — The lower square member of the base of a column, pedestal, or 
wall. 

Porch. — An exterior appendage to a building, forming a covered approach 
to one of its principal doorways. 

Portal. — The arch over a door or gate ; the framework of the gate ; the 
lesser gate, when there are two of difTerent dimensions at one entrance. 

Portcullis. — A strong timber gate to old castles, made to slide up and 
down vertically. 

Portico. — A colonnade supporting a shelter over a walk, or ambulatory. 

Priory. — A building similar in its constitution to a monastery or abbey, 
the head whereof was called a prior or prioress. 

Prism. — A solid bounded on the sides by parallelograms, and on the ends 
by polygonal figures in parallel planes. 

Prostyle. — A building with columns in front only. 

Pnrlines. — Those pieces of timber which lie under and at right angles to 
the rafters to prevent them from sinking. 

Pycnostyle. — An intercolumniation of one and a half diameters. 

Pyramid. — A solid body standing on a square, triangle, or pol)^gonal basis 
and terminating in a point at the top. 

Quarjy. — A place whence stones and slates are procured. 
Quay. — (Pronounced hey.) A bank formed towards the sea or on the side 
of a river for free passage, or for the purpose of unloading merchandise. 
Quoin. — An external angle. See Rustic (juoins. 

Rabbet, or Rebate. — A groove or channel in the edge of a board. 

Ramp. — A concave bend in the back of a hand-rail. 

Rampant arch. — One having abutments of different heights. 

Regula. — The band below the taenia in the Doric order. 

Riser. — In stairs, the vertical board forming the front of a step. 

Rostrum. — An elevated platform from which a speaker addresses an audi- 
ence. 

Rotunda. — A circular building. 

Rubble-wall. — A wall built of unhewn stone. 

Rudentiire. — The same as cable, which see. 

Rustic quoins. — The stones placed on the external angle of a building, pro- 
jecting beyond the face of the wall, and having their edges bevelled. 

Rustic-work. — A mode of building masonry wherein the faces of the stones 
are left rough, the sides only being wrought smooth where the union of the 
stones takes place. 

Salon, or Saloon. — A lofty and spacious apartment comprehending the 
height of two stories with two tiers of windows. 



636 APPENDIX. 

Sa7'cophagiis. — A tomb or coffin made of one stone. 

Scantl'uig. — The measure to which a piece of timber is to be or has been 
cut. 

Scarjitig. — The joining of two pieces of limber by bolting or nailing trans- 
versely together, so that the two appear but one. 

Scotia. — The hollow moulding in the base of a column, between the fillets 
of the tori. 

Sa'oll. — A carved curvilinear ornament, somewhat resembling in profile 
the turnings of a ram's horn. 

Sepulchre. — A grave, tomb, or place of interment. 

Sewer. — A drain or conduit for carrying off soil or water from any place. 

Shaft. — The cylindrical part between the base and the capital of a column. 

Shoar. — A piece of timber placed in an oblique direction to support a 
building or wall. 

.5"///. — The horizontal piece of timber at the bottom of framing ; the timber 
or stone at the bottom of doors and windows. 

Soffit. — The underside of an architrave, corona, etc. The underside of 
the heads of doors, windows, etc. 

Siiinmer. — The lintel of a door or window ; a beam tenoned into a girder 
to support the ends of joists on both sides of it. 

Systyle. — An intercolumniation of two diameters. 

Tccnia. — The fillet which separates the Doric frieze from the architrave. 

Talus. — The slope or inclination of a wall, among workmen called bat- 
tering. 

Terrace. — An area raised before a building, above the level of the ground, 
to serve as a walk. 

Tesselated pavement. — A curious pavement of mosaic work, composed of 
small square stones. 

Tetrastyle. — A building having four columns in front. 

Thatch. — A covering of straw or reeds used on the roofs of cottages, 
barns, etc. 

Theatre. — A building appropriated to the representation of dramatic 
spectacles. 

Tile. — A thin piece or plate of baked clay or other material used for the 
external covering of a roof. 

Tojub. — A grave, or place for the interment of a human body, including 
also any commemorative monument raised over such a place. 

Tortis. — A moulding of semi-circular profile used in the bases of col- 
umns. 

To'ver. — A lofty building of several stories, round or polygonal. 

Transept. — The transverse portion of a cruciform church. 

Transom. — The b?am across a double-lighted window ; if the window 
have no transom,, it is called a clere-story window. 

Thread. — That part of a step which is included between the face of its riser 
and that of the riser above. 

Tj'ellis. — A reticulated framing made of thin bars of wood for screens, win- 
dows, etc. 



GLOSSARY. 63 



TiiglypJt. — The vertical tablets in the Doric frieze, chamfered on the two 
vertical edges, and having two channels in the middle. 

Tripod. — A table or seat with three legs. 

Trochiliis. — The same as scotia, which see. 

Truss. — An arrangement of timbers for increasing the resistance to cross- 
strains, consisting of a tie, two struts, and a suspending-piece. 

Turret. — A small tower, often crowning the angle of a wall, etc. 

Ttisk. — A short projection under a tenon to increase its strength. 

Ty7npanum. — The naked face of a pediment, included between the level and 
the raking mouldings. 

Undcjpinning. — The wall under the ground-sills of a building. 

University. — An assemblage of colleges under the supervisionof a senate, etc. 

Vault. — A concave arched ceiling resting upon two opposite parallel walls. 

Venetian-door. — A door having side-lights. 

Venetian-window. — A window having three separate apertures. 

Veranda. — An awning. An open portico under the extended roof of a 
building. 

Vestibule. — An apartment which serves as a medium of communication to 
another room or series of rooms. 

Vestry. — An apartm.ent in a church, or attached to it, for the preservation 
of the sacred vestments and utensils. 

Villa. — A country-house for the residence of an opulent person. 

Vineiy.-—h. house for the cultivation of vines. 

Volute. — A spiral scroll, which forms the principal feature of the Ionic and 
the Composite capitals. 

Voussoirs. — Arch-stones. 

Wainscoting. — Wooden lining of walls, generally in panels. 

Water-table. — The stone covering to the projecting foundation or other walls 
of a building. 

Well. — The space occupied by a flight of stairs. The space left beyond the 
ends of the steps is called the well-hole. 

Wicket. — A small door made in a gate. 

Winders. — In stairs, steps not parallel to each other. 

Zophorus. — The same ?^s frieze, which see. 

Zystos. — Among the ancients, a portico of unusual length, commonly appro- 
priated to gymnastic exercises. 



638 



APPENDIX. 



TABLE OF SQUARES. CUBES. AND ROOTS. 









cFrom Iliittoi 


"s JMathcmali 


•K.) 






No. 


Square. 


Cube. 


[ Sq. Root. 


CiibeRoot.! 


No. 


Square. 


Cube. 


Sq. Root. 


CubeRwt. 


FT 


1 


1 


1-0000000 


roooooo! 


68 


46-24 


314432 


8-2462113 


4-0816551 


2 


4 


8 


14142136 


1-259921' 


69 


4761 


323509 


8 30662391 4 1015G6 


3 


9 


27 


l-73iJ()5U3 


1'442250 


70 


4900 


343000 


8-36660031 4-121285 


4 


16 


64 


2-0000000 


l'-537401 


71 


5041 


357911 


8-426 1498! 4-140318 


5 


25 


125 


2-2360r)80 


1-709976 


72 


5184 


373-248 


8-48528141 4-160168 


6 


36 


216 


2-44918^7 


1-817121 


73 


5329 


389017 


85440037 


4-179339 


7 


49 


343 


2-6457513 


1-912931 


74 


5476 


405224 


8-6023253 


4- 193336 


8 


64 


512 


2-8234271 


2000000 


75 


5625 


421375 


8-6602540 


4 217163 


9 


81 


729 


300000 JO 


2-080034 


76 


5776 


433976 


8-7177979 


4 2353-24 


10 


100 


1000 


3-1622777 


2-154435 


77 


5929 


456533 


8-7749644 


4-254321 


n 


121 


1331 


3-3l6')243 


2-223J30 


73 


6034 


474552 


8-8317609 


4-272659 


12 


144 


1723 


3-4641016 


2-2 -(9429 


79 


6241 


493039 


8-83319441 4-2-:HXS10| 


13 


169 


2197 


3 6,155513 


2 351335 


80 


6400 


512000 


8-944-2719 


4-303%9 


14 


196 


2744 


3-7416574i 2410142 


81 


6561 


531441 


9-tMH)0()00 


4-3j6"/49 


15 


225 


3375 


3 8729333 


2-466212 


82 


6724 


551368 


9-0553351 


4-344481 


16 


256 


40J6 


400J0000 


2-519342 


83 


6339 


571787 


9-1104336! 4-362071 


17 


239 


4913 


4- 123! 056 


2-571232 


84 


7056 


592704 


9-1651514 4-379519 


18 


324 


5332 


4-2426407 


2-6-2074 li 


85 


7225 


614125 


9-2195445 4396330 


19 


361 


6359 


■4-358_-i989 


2-66 -(402 


86 


7396 


63605:5 


9-2736185 4-414005 


20 


4L)0 


8000 


4-472 ISoO 


2-714418; 


87 


7569 


653503 


9-3273791 4-1,31048 


21 


441 


9261 


4-5825757 


2^758924: 


83 


7744 


681472 


9-3808315 4-4.:796[) 


22 


434 


10643 


4-6904153 


2-802039: 


89 


7921 


704969 


9-4339311 


4-164745 


23 


529 


12167 


4-7953315 


2-8433671 


90 


8100 


729000 


9-436333() 


4 -J 8 1405 


24 


576 


13321 


4-898.^795 


2-884499 


91 


8281 


753571 


9-5393920 


4-497941 


25 


625 


15625 


5-OJOOOOO 


2-924018 


92 


8464 


773633 


9-5916630 4 514357] 
9 64365081 4-530655| 


26 


676 


17576 


5-0990195 


2-962496, 


93 


8649 


804357 


27 


729 


19633 


5-1961524 


3000000' 


94 


8836 


8305 S4 


9-6953597 


4 516336 


23 


731 


21952 


5 2915:)2i') 


3036539 


95 


9025 


857375 


9-7467943 


4-562903 


29 


841 


24 38 J 


5-3351648 


3-0723 17i 


96 


9216 


834736 


9-7979590 


4-573357 


30 


900 


27000 


5-4772-25"i 


3-107232' 


97 


9409 


912673 


9-8483573 


4-594701 


31 


961 


29791 


5 5677644 


3141331! 


98 


9604 


941192 


9 8994949 


4-610435 


32 


1024 


32763 


5-6568542 


3-174802 


99 


9301 


970299 


9-9493744 


4-626065 


33 


1089 


35937 


5-74456-26 


3-2J7534I 


100 


10000 


lOOOOOO 


lO-OOOOOOOl 4-641539 


34 


1156 


393J4 


5-8309519 


323J612 


101 


10201 


103)301 


10-0498755 4-657009 


35 


1225 


42375 


5-9160798 


3-271066 


102 


10404 


1061208 


10-099504'): 4 '072329 


36 


1296 


4665() 


6OOO0O0O 


3 3J1927 


103 


10609 


1092727 


10- 14839 iO; 4-G37548 


37 


1369 


50653 


60S27625 


3-332222 


104 


10816 


1124864 


10-1 9803- io! 4-702659 


• 33 


1444 


54372 


6-1644140 


3361975 


105 


1 1025 


1157625 


10-24695 J;3" 4-717694 


39 


1521 


59319 


6-2449980 


3-3J1211; 


106 


11236 


1191016 


10-29553D-J 4-732623 


40 


1600 


640J0 


6-3245553 


3 419952 


107 


11449 


1 •2-25043 


10-3440301] 4-747459 


41 


, 1681 


63921 


6-4031242 


3448217 


103 


11664 


1259712 


10-392334.3; 4762203 


\i 


1764 


74038 


6-4307407 


3-476027 


109 


11831 


1295029 


10-4403J65' 4-776856 


43 


1819 


79507 


6-5574335 


3503393 


110 


12100 


1331000 


10-48308-^5 4-791420 


44 


1936 


85181 


6 6332496 


3-53)313 


111 


12321 


1367631 


10-535r5:iSl 4-305895 


45 


20^5 


91125 


6-7032039 


3556393 


112 


12514 


1404923 


10-5330052! 4-a20284 


46 


2116 


9733C) 


6-7323300 


3-533948 


113 


12769 


144-2397 


10-6301453' 4-834588 


47 


2209 


103 S23 


6-8556546 


3-60S326 


114 


1-2996 


1431544 


10-6770733' 4-848308 


43 


23i)4 


1 10592 


692S2032 


3 634-241 


115 


13225 


1520375 


10 7-233i;53i 4-862944 


49 


2401 


117r)49 


70000000 


3-6593J6 


116 


13456 


1560396 


10-77032.V:! 4-876999 


50 


2500 


125000 


7-0710678 


3 6-!403l 


117 


13639 


1601613 


10-816:!53>^i 4-890973 


51 


2601 


132J51 


7- 14 14234 


3-70343.) 


118 


13924 


1643032 


10-86-278()5! 4-90486S 


52 


2704 


140i>03 


7-2111026 


3-732511 


119 


14161 


1685159 


10-9037 12 ij 4-913685 


53 


231)9 


143377 


7-2^010J9 


3-756236 


120 


14400 


1723000 


10-954151"! 4-93 i 124 


54 


2916 


157464 


7-3181692 


3-779763 


121 


14641 


1771561 


ll-OO0O-:>:>;; 


4-946087 


55 


3025 


166375 


7-4161935 


3-S02952 


1-22 


14384 


1815348 


ll-0453i>l(; 


4-959676 


56 


3136 


175616 


7-4833143 


3-8258.i2 


123 


15129 


1860367 


11-0905365 


4-973190 


57 


3^19 


185193 


7-5193341 


3-34S50II 


121 


15376 


1906624 


11- 1355-237 


4-936631 


58 


3364 


195112 


7-6157/31 


3 •8703771 


125 


15625 


19531-25 


11-180339-J 


5-000000 


59 


3431 


205379 


7-6311457 


3-892996' 


126 


15376 


2000376 


11-2249722 


5-013293 


60 


3(J00 


2 1600 J 


7-7459667 


3-914863] 


127 


16129 


2048333 


11-2694277 


5-026526 


61 


3721 


226931 


7-8102197 


39364J7| 


123 


16334 


2097152 


11-3137085 


5-039684 


62 


3S41 


23S323 


7-8740079 


3-957891 


129 


16611 


2146639 


11 -3573! 6? 


5 052774 


( 63 

|64 


3969 


250017 


7-9372539 


3-979057 


130 


16900 


2197000 


114017543 5 065797 


4iiy(J 


262144 


8 0000000 


4-000000 


131 


17L61 


2-243091 


11-4455231 5-078753 


65 


4225 


274625 


8-0622577 


4-020726! 


132 


17421 


2299963 


11-4391253 5-091643 


«■)(■) 


435f) 


2374 'j6 


8-12403^4 


4-041240 


133 


17689 


235:^637 


11-53256261 5 KM 469 


!'■'" 


4»89 


300763 


8-l«5352i 


4061548 


134 


17956 


24i)6|04 


1I-575S369! 5 117230 



TABLE OF SQUARES, CUBES, AND ROOTS. 



6;g 



No. 


Sqiiiirc. 


Cube. 


S(l. Root. 


CubeRootJ 

i 


i\0. 


Square. 


Cube. 


Sq. Root. 


CubcRool. 


135 


18225 


2460375 


11-6189500 


5-129928: 


202 


40804 


824-2408 


14-21-26704 


5-867461 


136 


18496 


2515456 


11-661903S 


5-142563 


203 


41209 


83654-27 


14-2478088 


5-877131 


137 


13769 


2571353 


11-7046999 


5-155137i 


204 


41616 


8489664 


14-2323569 


5-836765 


133 


19044 


2628072 


11-7473401 


5-1676491 


205 


42025 


8615125 


14-3178211 


5-896368 


139 


19321 


2685619 


11-7898261 


5-1801011 


206 


42l3f. 


8741816 


14.35-27001 


5-905941 


140 


19600 


2744000 


11-8321598 


5-192494 


207 


42849 


8869743 


14-3374946 


5-915432 


141 


19881 


2803221 


11-8743422 


5-204828: 


203 


43-264 


8998912 


lt-4222051 


5-924902 


142 


20164 


2S63283 


11-9163753 


5-217103 


209 


43881 


9129329 


14-4568323 


5-934473 


143 


20449 


2924207 


11-9582607 


5-2-29321: 


210 


44100 


9261000 


14-4913767 


5-943022 


144 


20736 


2985984 


12-000()000 


5-241483 


211 


44521 


9393931 


14-5258390 


5-953342 


145 


21025 


3048625 


12-0415946 


5-253533 


212 


44944 


95-28123 


14-5602198 


5-96-2732 


146 


21316 


3112136 


12-0830460 


5-265637 


213 


45369 


9663597 


14-59451951 5-97-2093| 


147 


21609 


3176523 


12-1243557 


5-277632 


214 


45796 


9800344 


14-6-287333 


5-931424 


148 


21904 


3241792 


12-1655-251 


5-239572 


215 


46i25 


9933375 


14-682S783 


59S0726 


149 


22201 


3307949 


12-2065558 


5 301459 


216 


46656 


10077696 


14-6969335 


6-OOOOOU 


150 


22500 


3375000 


12-2474487 


5-313203 


217 


47089 


10218313 


14-7309199 


6-000245 


151 


22,S01 


3442951 


12-2832057 


5-3-25074 


218 


47524 


10.380232 


14-7648231 


6-013462 


152 


23104 


3511808 


12-3288280 


5-336803 


219 


47961 


10503459 


14-7986486 


6-027650 


153 


23409 


3581577 


12-3693169 


5-348481 


220 


48400 


10648000 


14-8323970 


6-036811 


154 


23716 


3652264 


12-4096738 


5-380103 


221 


48S41 


10793381 


14-8660687 


6045943 


155 


24025 


3723875 


12-449899.. 


5-371685 


222 


49234 


1U941048 


14-8906644 


6-055049 


156 


24336 


3796416 


12-4399960 


5-333213 


223 


49729 


11030567 


14-9331845 


6-064127 


157 


24619 


3869393 


12-5299641 


5-394691J 


224 


50176 


11239424 


149638295 


6073178 


158 


24964 


3944312 


12 •5698051 


5-406120 


225 


506-25 


113906-25 


15-0000000 


6-082202 


159 


25281 


4019679 


12-6095202 


5-417501 


226 


51076 


11543176 


15-0332964 


6-091 19y 


160 


25600 


4096000 


12-6491106 


5-428835: 


227 


51529 


11897083 


150865192 


6-100170 


161 


25921 


4173281 


12-6385775 


5-440122 


223 


51934 


11852352 


15-0996639 


6109115 


162 


25244 


4251528 


12-7279221 


5-451332 


229 


52441 


12008939 


151327460 


6-113033 


163 


26569 


4330747 


12-7671453 


5-462556 


230 


5290U 


12167000 


15- 1657509 


6-126928 


16i 


!i6896 


4410944 


12-8062485 


5-473704! 


231 


53361 


123-28391 


15 1936342 


6-135702 


165 


27225 


4492125 


12-8452326 


5-434807 


232 


53824 


12487168 


15-2315 462 


6-144634 


166 


£7558 


4574296 


12-8840987 


5-495365 


233 


54-389 


12649337 


15-2643375 


6-153449 


167 


27839 


4657463 


12 9-228480 


5-506878 


234 


54758 


1281-2904 


15 2070585 


6-162210' 


168 


2S224 


4741632 


12-9614814 


5-517848 


235 


55225 


12977875 


15-3207097 


6-171008 


169 


2^561 


4826809 


130000000 


5-528775' 


235 


55896 


13144256 


15 3322015 


6-179747 


170 


28900 


4913000 


13 0384048 


5-539658; 


237 


56169 


1331-2053 


15-3J43043 


6- 183463 


171 


29211 


5000211 


13-0766968 


5-550490: 


233 


38644 


13481272 


15-4272486 


6-197154 


172 


29584 


5088448 


13-1148770 


5-581293 


239 


57121 


13851919 


15-4506248 


6-205822 


173 


29929 


5177717 


13-1529464 


5-572055 


240 


57600 


13324000 


15-4919334 


6214465 


174 


30276 


5268024 


13-1909060 


5 532770; 


241 


53081 


139.*7521 


15-5241747 


8-223034 


175 


30525 


5359375 


13-2-287568 


5-593445! 


242 


58564 


14172433 


15 5583492 


8-231630 


176 


30976 


5451776 


13-2664992 


5-604079 


243 


59049 


14348907 


15-5334573 


6-240251 


177 


3132'J 


5545233 


13-3041347 


5-614672' 


244 


39336 


14526784 


15-6-204994 


6-24S3O0 


178 


31(:3i 


5639752 


13-3416341 


5 •625-226 


245 


60023 


147061-25 


15-6524753 


6-257325 


179 


32041 


5735339 


13-3790882 


5-635741! 


246 


60516 


14836936 


15-6343S71 


6-285327 


180 


32400 


5832000 


13-4164079 


5-646216! 


247 


61009 


15089223 


15-7162338 


6-274305 


181 


32751 


5929741 


13-4536240 


5-656653! 


248 


61504 


15252902 


15-7480157 


8-232761 


182 


33 121 


6028568 


13 4907376 


5-667051 


249 


6-2001 


15433249 


15-7797.333 


6-291195 


183 


33439 


6128487 


13-5277493 


5-677411 


250 


62500 


15825000 


15-8113-!33 


8-200605 


181 


33356 


6229504 


13-5648600 


5-637734 


251 


63001 


15313251 


15-84-29795 


6-307994 


185 


34225 


6331625 


13-6014705 


5-693019 


252 


63504 


16003008 


158745079 


6-316360 


18^\ 


34596 


6434856 


13-6331817 


5-708267| 


253 


64009 


18194277 


15-9059737 


6-324704 


187 


34989 


6539203 


13-6747943 


5-718479 


254 


64516 


18337064 


15-9373775 


6-333026 


188 


35344 


6644672 


13-7113092 


5-7238541 


255 


65025 


16531375 


15 9687194 


6-341323 


189 


35721 


6751269 


13-7477271 


5-733794 


256 


65536 


16777216 


16 0000000 


6-349604 


190 


36100 


6859000 


13-7840483 


5-748397] 


257 


68049 


16074593 


16 0312195 


6357861 


191 


36481 


6967871 


13-8202750 


5-758965| 


253 


66564 


17173512 


16-0623784 


6-366007 


192 


36384 


7077838 


13-8564065 


5-763998 
5-778996 


259 


67081 


17373079 


16-0034760 


6 374311 


193 


372 39 


7189057 


13-8924 440 


260 


67600 


17576000 


16-1245155 


6-382504 


194 


37,'.36 


7301384 


13-9283383 


5-788960 


261 


63121 


17779581 


1615.34044 


8-300676 


195 


38025 


7414875 


13-964-2400 


5-798390 


262 


6S644 


17984723 


18-1864141 


8-398323 


196 


33416 


7529536 


14-0000000 


5-808736 


283 


69169 


18191447 


16-217-2747 


8-406953 


197 


38809 


7645373 


14-0356683 


5-818648 


264 


60696 


18399744 


18-2430763 


6-415069 


198 


39204 


7762392 


14-0712473 


5-823477 


263 


702-25 


18609625 


18-27d8206 


8-423 15-^ 


199 


39n01 


7880599 


141067360 


5-833272 


268 


70756 


18821098 


16-3005064 


6-43122- 


2<W 


10000 


8000000 


14-1421356 


5-848035 


267 


71230 


19034163 


16-3401346 


6-439277 


201 


40J01 


8120601 


14-1774469 


5-857766 


268 


71824 


19248832 


16-370m)55 


6-447306 



640 



APPENDIX. 



Na 


Square. 


Cube. 


Sq. Root. 


CubeRootJ 


No. 


Square. 


Cube. 


Sq Hoot. |cubeU"oi. 


269 


72351 


19465109 


16-4012195 


6-455315 


336 


1 12896 


37933056 


18-33030281 6-952053 


270 


72900 


19633000 


16-4316767 


6-463304 


337 


113559 


33272753 


18 357559Si 6-958943 


271 


73441 


19902511 


16-4620776 


6-471274 


333 


114244 


33514472 


18-3347763 6-9658-20 


272 


73984 


20123648 


16-49-24225 


6-479224 


339 


114921 


389['.8219 


18-41195-26 


6-97-2683 


273 


74529 


20346417 


16-5227116 


6-487154 


340 


115600 


393f>4000 


18-4390889 


6-979532 


274 


75076 


20570824 


16'55-29454 


6-495065 


341 


116281 


39651821 


184661853 


6-986368 


275 


75025 


20796875 


16-5831240 


6-502957 


342 


116964 


40001688 


18-49324-20 


6-993191 


276 


76176 


21024576 


16-6132477 


6-510830 


342 


117649 


40353607 


18-5202592 


7-000000 


277 


76729 


212539.33 


16-6433170 


6-518634 


344 


1 18336 


40707584 


18-5472370 


7-006796 


278 


77284 


21484952 


16-6733320 


6-526519, 


345 


119025 


410636-25 


18-5741756 


7-013579 


279 


77811 


21717633 


16-7032931 


6-534335 


346 


119716 


41421736 


18-6010752 


7-020349 


230 


78400 


21952000 


16-7332005 


6-5121331 ^47 


120409 


41781923 


18-6279360 


7-027106 


281 


78961 


22188041 


16-7630546 


6-549912 


343 


121104 


42144192 


18-6547581 


7-033850 


282 


79524 


22425763 


16-7923556 


6-557672 


349 


121801 


42508549 


18-6815417 


7-040581 


283 


80089 


22665187 


16-82-26033 


6-555414 


350 


122500 


42875000 


18-7082869 


7-047299 


284 


80656 


229063M 


16-85-22995 


6-573139 


351 


123201 


43243551 


18-7349940 


7-054004 


235 


81225 


23149125 


16-8819430 
169115345 


6-580344 


352 


123904 


43614-208 


18-7616530 


7-060697 


286 


81796 


23393656 


6-533532: 353 


124609 


43985977 


18-7832942 70673771 


237 


82369 


23639903 


16-94107431 6-596202! 


354 


125316 


44361864 


18-8148877 


7-074044 


283 


82944 


23387872 


16-9705627 


6-603354 


355 


126025 


44738875 


18-8414437 


7-080699 


239 


83521 


24137569 


17-0000000 


6-611489 


356 


126736 


45118016 


18-8679623 


7-087a41 


290 


84100 


24389000 


17-0293864 


6-619106 


357 


127449 


45499-293 


18-8944436 


7-093971 


291 


84681 


24642171 


17-0537221 


6-626705; 358 


128164 


4538-2712 


18-9208379 


7-100583 


292 


85264 


24897083 


17-0880075 


6-634237j 359 
6 641852^1 360 


128881 


46268279 


18-947-2953 


7-107194 


293 


85849 


25153757 


17-1172428 


129600 


46656000 


18-9736660 


7-113787 


294 


86136 


25412184 


17-1464232 


6-6494001 361 


130321 


47045331 


19-0000000 


7-1-20367 


295 


87025 


25672375 


17-1755640 


6-656930 352 


131044 


47437928 


19-0262976 


7-1-26935 


296 


87616 


25934336 


17-2046505 


6-664444 353 


131769 


47832147 


19-05-25589 


7-133492 


297 


83209 


25198073 


17-2335379 


6-671940 354 


132495 


48223544 


19-0737840 


7-140037 


293 


83304 


26463592 


17-2626765 


6-679420' 1 355 


13.3225 


436271-25 


19-1049732 


7-146569 


299 


89401 


26730899 


17-2916165 


6-635383 • 366 


133956 


49027395 


19-1311265! 7 1530931 


300 


90000 


2700000-0 


17-3205081 


6-694329 367 


134639 


49430863 


19-1572441 7-1595991 


301 


90601 


27270901 


17-3493516 


6-701759 368 


135424 


49836032 


19-1833-261 


7-166096 


302 


91204 


27543603 


17-3731472 


6-7091731 369 


136161 


50243409 


19 20937-27 


7-17-2581 


303 


91809 


27818127 


17-4068952 


6-716570![ 370 


136900 


50653000 


19-2353341 


7-179054 


304 


92416 


28094464 


17-4355958 


6-723951 [I 371 


137641 


51064811 


19-2613503 


7-185516 


305 


93025 


28372625 


17-4642492 


6-7313l6j| 372 


138334 


51478848 


19-23730151 7-1919661 


336 


93636 


23652616 


17-4928557 


6-733664!! 373 


139129 


51895117 


19-3132079 


7-198405 


307 


94249 


23934443 


17-5214155 


6-7459971 374 


139876 


52313624 


19-3390796 


7-204832 


308 


94864 


29218112 


17-5499288 


6-753313, 375 


140625 


52734375 


19-3649167 


7-211248 


309 


95481 


29503629 


17-5783958 


6-760614 376 


141376 


53157376 


19-3907194 


7-217652 


310 


96100 


29791000 


17-6068169 


6-767899 37T 


142129 


53582633 


19-4164878 


7-224045 


311 


96721 


30080231 


17-6351921 


6-775169 3:8 


142834 


54010152 


19-442-2221 


7-230427 


312 


97344 


30371328 


17-6635217 


6-782423'! 379 


143541 


54439939 


19-4679223 


7-236797 


313 


97969 


30664297 


17-6918060 


6-789661 ! 330 


144400 


54872000 


19-4935337 


7-243156 


3H 


98596 


30959144 


17-7200451 


6-796834! 331 


145161 


55306341 


195192213 7-249504 


315 


99225 


31255375 


17-7432393 


6-804092 332 


145924 


5574-2953 


19-5443-203: 7255341 


3Iii 


99856 


31554496 


17-7763333 


6-811235' 333 


146639 


56181837 


I9-5703353I 7-262167 


317 


100489 


31855013 


17-8044933 


6-818462, 334 


147456 


55623104 


19-5959179| 7-263432 


318 


101124 


32157432 


17-8325545 


6-8-256-24 


335 


148-225 


57066625 


19-6214169, 7-274786 


319 


101761 


32461759 


17-8605711 


6-83-2771 


336 


148996 


5751-2456 


19-64633-27, 7231079 


320 


102400 


32763000 


17-8835433 


6-839904 


337 


149769 


57960603 


1967-23156 


7-237362 


321 


103041 


33076161 


17-9164729 


6-847021 


333 


150544 


58411072 


19-6977156 


7-293633 


322 


103584 


33336248 


17-9443534 


6854124 


339 


151321 


53363859 


19-7-230829 


7-299894 


323 


104329 


33598267 


17-97-22 J08 


6-861212 


390 


152100 


59319000 


19-7434177 


7-306144 


324 


104976 


34012224 


18-0000000 


6-858235 


3J1 


15-2831 


59776471 


19-7737199 


7 312333 


325 


105625 


34323125 


18-0277564 


6-875344 392 


153564 


60236233 


19-7939399 


7-318611 


326 


106276 


34645976 


18-0554701 


6-832339 


393 


154449 


60693457 


19-8242276 


7-3248-29 


327 


10C>929 


34965733 


180331413 


6-889419 


394 


155-236 


6116-2984 


19-8494332 


7 331037 


323 


107584 


35287552 


18-1107703 


6-896435 


395 


1-.6025 


61629375 


19-8746069 


7-337-234 


329 


108241 


3561 12S9 


181333571 


6 903436 


396 


15'^816 


, 6-2099135 


19-8997487 


7-3434-20 


3311 


108900 


35937000 


181659021 


6-910423 


397 


157609 


62570773 


19-9243533 


7-349597 


331 


109561 


36264691 


18-1934054 


6-917396 


308 


158404 


63044792 


19-9499373 


7-355762 


332 


110224 


36594368 


18-2208672 


6-924356 


399 


159201 


63521199 


19-9749844 


7361918 


333 


110839 


36926037 


18-2482376 


6-931301 


400 


160000 


64000000 


20-0000000 


7-363063 


:^3i 


111556 


37259704 


18-2756669 


6-933232 


! 401 


160801 


64431201 


20 0249344 


7-374193 


335 


112225 


37595375 


18-3030052 


6-945150 


402 


161604 


1 64954303 


20*^99377 


7-330323 



TABLE OF SQUARES, CUBES, AND ROOTS. 



641 



No. Square. 



S(i. Root. CubeRoot. No. I Squar 



431 



441 



444 
44.0 
446 
447 

44S 



1971361 
198025; 
1939161 
199809; 
200704' 
449; 20 160 1; 

450 202500 

451 203401 
452| 204304 
453 21)5209 
4541 206116 
453 207025 
456i 207936 
4571 208849 
458| 209764 
4591 210681 
460! 211600 
461 212521 
4621 213444 
463! 21436i) 
464| 215296 
465! 2 i 6225 
466| 217156 
46? 218:)89 

468 2190241 

469 2199611 



65450327 
?5939264 
C643J125I 
669231161 
67419143 
67917312 
68417929 
68921000 
69426/31 
699341 li 
70444i}97 
70J57944 
71173375 
71991296 
72511713 
73034632 
73560059 
74038000 
74618461 
75151448 
75636967 
76225024 
76765625 
773037761 
77854483 
78402752 
78953589 
79507000 
80082991 
80621568 
811827371 
81746504 
82312875' 
82881856 
83453453 
84027672 
846015191 
85184000' 
8576 •) 121 1 
86350388 
869383071 
87528384' 
83121125 
8871653<; 
89314623 
89915392 
90518319 
91125000 
91733851 
92345403; 
929598771 
93576664 
94196375 
94818816' 
95443993 
96071912' 
96702579 
97336000 
97972181 
98:")11123 
99252847| 
99897344^ 
100514625 
101194696 
101817563 
102503232 
103161709 



20-0748599 
20091)7512 
201246118 
20-1494417 
20-1742410 
20-1990099 
2J-2237484 
20-2484567 
20-2731349 
20-2977831 
20-32-24014 
20-3469899 
20-3715488 
20-3960781 
20-4205779 
20-4450483 
20-46;)4395 
20-4939015 
20-5182845 
20-5426336 
20-5669638 
20-5912603 
20-6155281 
20-6397674 
20-6633783 
20-6831609 
20-7123152 
20-7364414 
20-7605395 
20-7846097 
20-8086520 
20-8326667 
20-856653O 
20-8806130 
20-9045450 
20-9284495 
20-9523268 
20-9761770 
21-00000001 
21-0237960 
21-0475652' 
21-0713075; 
21-09502311 
21-11871211 
211423745 
21-1660105 
21-1896201 
21-2132034 
21-2367606 
21-2802916 
21-28379671 
21-3072758! 
21-3307290' 
21-35415551 
21 ■3775533 
21-40093461 
21-4242353 
21-4476106 
21-4709106 
21-4941853 
21-5174318! 
21-54065921 
21-56335871 
21-5370331 
21-6101823 
21-6333077 
, 21-6564078 



7-3S6437 

7-392542 

7-398636 

7-404721 

7-410795 

7-416859 

7-422914 

7-428959 

7-434994 

7-441019 

7-447034 

7-453040 

7-459033 

7-465022 

7-470999 

7-476966 

7-4829-24 

7-488872 

7-494311 

7-500741 

7-506661 

7-5 12571 

7-518473 

7-524365 

7-530243 

7-538122! 

7-541987' 

7-547842 

7-553639 

7-559526 

7 585355 

7-571174 

7-576985 

7-532786 

7-583579 

7-5J4363 

7600133 

7-()05905 

7-611663 

7-617412 

7-6-23152:! 510i 

7-6233841I 511 

7-634607ii 512 

7 640321! 

7-646027| 

7-C51725! 

7-657414; 

7-663094; 

7-663766! 518 

7-6744301' 519 



470 



Cube. 



471 
472 
473 
474 

475 
476 
477 

478 
479 
4i0 
4S1 
4S2 
4S3 
4S4 
435 
486 
437 
438 
439 
490 
491 
492 
493 
1 494 
495 
496 
497 
493 
499 
500 
501 
502 
503 
504 
505 
506 
507 
508 
509 



513 
514 
515 
516 
517 



-680086 520 



7-685733:1 521 
7-691372i| 522 
7-6J70021i 
7-702625; I 
7-7082391 j 
7-713345: 
7-719443'| 
7-725032; 1 
7-7306 14 i! 
7-736183:1 530 
7-7417531 531 
7-747311^ 5-32 
7-75-286 ill 533 
7-7584021 1 531 
7-763j36i 535 
7-769462; I 536 



523 
5-24 
525 
526 
5-27 
523 
5-29 



2209001 

221841 

222784 

223729 

2-24676 

225825 

226576 

227529 

223484 

2-29441 

230400 

231361 

232324 

233289 

234256 

2352-25 

236196 

237169 

238144 

239121 

240100 

241031 

212064 

243049 

244036 

245025 

246016 

247009 

243004 

249001 

250000 

251001 

252004 

253009 

254016 

255025 

256036 

257049 

253064 

259031 

260100 

261121 

262144 

263169 

264196 

265225 

266256 

267289 

263324 

269331 

270400 

271441 

272434 

273529 

274576 

275625 

276676 

277729 

278734 

279341 

230900 

281961 

283024] 

234089 

•285156' 

236-225' 

287206 



Sq. Root. CubeRooi 



103323000 

104487111 

105154048 

105823817 

106496424 

107^71875 

107850176 

108531333 

109215352 

109902239 

110592000 

111-284641 

111930163 

11-2678537 

113379004 

1140841-25 

114791256 

115501303 

116214272 

116930169 

117649000 

118370771 

119005483 

119823157 

1-20553784 

121287375 

122023036 

122763473 

123505902 

1-24251499 

125000000 

125751501 

126506008 

127-263527 

1-28024064 

128787625 

129554216 

130323843 

131096512 

131872-229 

13-2651000 

133432331 

134217728 

135005697 

135796744 

136500875 

137383098 

133183413 

138991832 

139798359 

14060800U 

141420761 

14-2236648 

143055667 

143377824 

144703125 

145531576 

146363183 

147197952 

148035889 

148877000 

149721291 

150588763 

151419437 

152-273304 

153130375; 

153990656 



21-6794334 

21-70-25344 

21-7-255610 

21-7485632 

21-7715411 

21-7914947 

21-8174242 

21-8403207 

21-8632111 

21-8880836 

21 -9080023 

21-93171-22 

21-9544934 

21-9772610 

22-OOOuOOO 

22-0227155 

22 0454077 

22-0680765 

22-0907220 

2-2-1133444 

22- 1359438 

2-2-1585193 

22- 181073 j! 

22-2038033| 

22-228110^ 

22-2485055 

22-2710575 

22-20349631 

22-3159138 

22-3333079 

22-3608798 

22-3330-293 

22-4053565 

22-4276615 

22-4499443 

22-4722051 

22-4944438 

22-5166805 

22-533^553 

22-5810283 

22-53317981 

22-6053091 

22-6274 170| 

22-6195033 

22-67158311 

22-6936 1141 

2271563341 

22-7376340; 

22-75081341 

22-7815715 

22-8033035' 

22-8254244' 

22-84731'j3 

22-8891933 

22-8010463 

22-91-23785' 

22-93 16 V09 

22-95,i48;6 

22-9732508 

23 0000000 

23-021728J1 

230434372 

•23-065 12 J 2' 

23-0887923 

231084400 

23-1300870 

23-1516738 



7-7749 

7-730490 

7-785993 

7-791487 

7-798974 

7-802454 

7-8079-25 

7-813389 

7-813846 

7 -82 4-294 

7-829735 

7-835169 

7 840595 
7-846013 
7-851424 
7-356823 
7-832'224 
7-867613 
7-872994 
7-878368 
7-833735 
7-839095 
7-894447 
7-899792 
7-9051-29 
7-910460 
7-915783 
7-92109i) 
7-928403 
7-931710 
7-937005 
7-94-2293 
7-947574 
7-952848 
7-953114 
7-963374 
7-968627 
7-973373 
7-97911 
7-934314 
7-989570 
7-994788 

8 000001 
8-0052(1: 
8-0L>4o:« 
8-015595 
8-020779 
8-025957 
8-0311-29 
803820/ 
8-041451 
8-016803 
8-051748 
8 058836 
806201b 
8067143 
8-072262 
8077374 
8 08-2430 
8 037579 
8-09267^ 
8-097759 
8-l0233y 
8-107913 
8-11-2930 
8-118041 

, 8-12309; 



642 



APPENDIX. 



No. 
"537 


Square. 


Cube. 


Sq. Root. 


CuheRoot. 


No. 


Square. 


Cube. 


Sq. Rcct. CuleRoou 


283369 


154854153 


231732605 


8-123145 


604 


3.'>4816 


220348864 


24-5764115 8 45302^ 


538 


2'!9444 


155720872 


231948270 


8-1331371 


605 


3(16025 


221445125 


24-5967478 8-457691 


539 


293521 


156590819 


23-2163735 


8-13S223! 


606 


3'17236 


222545016 


24-6170673 8-462348 


540 291600 


157464000 23-2379001 


8-143-253i 


607 


368449 


223643543 24-63r3700! 8-4670O0| 


541 292681 


158340421 23-2594067 


8-148276! 


608 


369664 


224755712 


24-6576560 8-471647 


542 


293764 


159220083 


23-2.S03935 


8-153294 


609 


370831 


225865529 


24-6779254 8-476289 


543 


294849 


160103007 


23-3023604 


8-153305 


610 


372100 


226981000 


246981781 8-480926 


544 


295936 


160939184 23-323S076 


8163310 


611 


373321 


223099131 


24-7184142 8-485558 


545 


297025 


161873625 23-345-2351 


8-168309 


612 


374554 


229220928 


24-7385338 8-490185 


546 


298116 


162771336 


23-3665429 


8-173302 


613 


3757fl9 


230346397 


24-7538358 


8-494806 


547 


299209 


163667323 


23-33303111 8-1782391 


611 


376996 


231475544 


24-7790234 


8-499423 


548 


300304 


164566592 


23-4093998 


8-183269 


615 


373225 


232608375 


24-7991935 


8-504035 


549 


301401 


165469149 


23-4307490 


•8-188244 


616 


379456 


233744895 


24-81934-73 


8-508642 


550 


302500 


166375000 


23-4520788 


8-193213 


617 


330689 


234885113 


24-8394347 


8-513243 


551 


3J3601 


167234151 


23-4733392 


8198175 


618 


331924 


235029032 


24-8596053 


8-517840 


552 


304704 


163196608 


23-4946302 


8-203132 


619 


333161 


237176659 


24-8797106 


8-522432 


553 


335809 


16911237.7 


23-5159520 


8-203032 


620 


331400 


233328060 


24-8997992 


8-527019 


554 


306916 


170031464 


23-5372046 


8-213027 


631 


335641 


239483061 


24-9198716 


8-531601 


555 


303025 


170953375 


23 5534330 


8-217966 


G32 


386384 


240641848 


24-9399278 


8-536178 


556 


309136 


171879616 


23-5796522 


8-2-32893 


623 


38^129 


241804367 


24-959b679 


8-540750 


557 


310249 


172808693 


23-6003474 


8-227825 


634 


339376 


242970624 


24-9799920 


8-545317 


558 


311.364 


173741112 


23 6220236 


8-232746 


625 


390825 


244140625 


25-0000000 


8-549880 


559 


312431 


174676379 


23-6431808 


8-237661 


626 


391876 


245314376 


25-0199920 


8-554437 


560 


313600 


175616000 


23-6643191 


8-242571 


627 


393129 


246491833 


25-0399681 


8-558990 


561 


314721 


176558481 


23-6854336 


8-247474 


623 


394334 


247673152 


25-0599282 


8-563538 


562 


315344 


177504328 


23-7065392 


8-252371 


629 


395641 


248853189 


25-0798724 


8-568081 


563 


316969 


178453547 


23-7276210 


8-257263 


630 


396900 


250047000 


250993003 


8-572619 


564 


318096 


179406144 


23-7486842 


8-262149 


631 


393161 


251239591 


25-1197134 


8-577152 


565 


319225 


180362125 


23-7697283 


8-267029 


632 


399424 


252435958 


25-1395102 


8-581681 


566 


320356 


181321496 


23-7907515 


8-271904 


633 


400689 


253636137 


25-1594913 


8-586205 


567 


321489 


182284263 


23-8117618 


8-276773 


634 


401956 


254840104 


25-1793566 


8-590724 


558 


322624 


183250432 


23-8327536 


8-281635 


635 


4U3225 


256047375 


25-1992063 


8-595233 


569 


323761 


184220009 


23-8537209 


8-236493 


636 


404196 


257259456 


25-2190404 


8-599748 


570 


324900 


185193000 


23-8746723 


8-291344 


637 


405769 


258474853 


25 2333539 


8-604252 


571 


326041 


186169411 


23-8956063 


8-2.)6190 


633 


407044 


259694072 


25-2585619 


8-608753 


572 


327184 


187149248 


23-9165215 


8-331033 


639 


408331 


260917119 


25 2734493 


8-613248 


573 


333329 


183133517 


23-9374184 


8-335365 


640 


409600 


262144000 


25.2932213 


8-617739 


574 


329476 


189119224 


23-9532971 


8-310694 


641 


410381 


263374721 


25-3179778 


8-622225 


575 


330625 


190109375 


23-9791576 


8-31551? 


642 


412164 


264609283 


25-3377189 


8-626706 


576 


331776 


191102976 


24-0000000 


8-32'33;^5 


643 


413449 


265847707 


25-3574447 


8-631183 


577 


332929 


192100033 


24-0208243 


8-325147 


644 


414735 


267089984 


25-3771551 


8-635655 


578 


334034 


193100552 


24-0416306 


8-329954 


645 


416025 


253336125 


25-3968502 


8.640123 


579 


335241 


194104539 


240524188 


8-334755 


646 


417316 


269586136 


25-4165301 


8-644585 


580 


336400 


195112000 


24-0831891 


8-339551 


! 647 


418509 


270340023 


25-4351947 


8-649044 


581 


337561 


196122941 


241039416 


8-344341 


! 648 


419904 


272097792 


25-4553441 


8-653497 


582 


333724 


197137368 


24-1246762 


8-349126|^ 649 


421-201 


273359449 


254754734 


8-657946 


583 


339339 


193155237 


241453929 


8-353905 i 650 


422500 


274625000 


25-4950976 


8-662391 


584 


a41056 


199176704 


24-1660919 


8 -353678! i 651 


423301 


275894451 


25-5147016 


8-666331 


585 


343225 


200201625 


24-1867732 


8-353447 


f 652 


425104 


2771678081 25-5342907 


8-671266 


586 


343396 


201230055 


24-2374369 


8-353-309 


653 


425409 


273445077] 25-5533647 


8-675697 


587 


344569 


202262003 


24-2230829 


8-3-72967 


, 654 


4-27716 


2797262641 25-5734237 


8-680124 


588 


345744 


203297472 


24 2487113 


8-377719 


1 655 


429035 


281011375! 25-5929578 


8-684546 


589 


346921 


204336469 


24-2593222 


8-332465 


: 656 


43.336 


282300416 25-6124969 


8-688963 


590 


348100 


205379000 


24-2899156 


8-337206 


i 657 


431649 


283593393 25-6330112 


8-693376 


591 


349281 


1 206425071 


24-3104916 


8-391942 


I 653 


432964 


2348903121 25-6515107 


8-697784 


•^92 


350464 


1 207474688 


24-3310501 


8-396673 


1 659 


434281 


2361911791 25-6709953 


8-702188 


593 


351649 


1 208527857 


24-3515913 


8-4013981 660 
8-406 Uh! 661 


435600 


237496000! 25-6904652 


8-706538 


594 


352836 


; 209584584 


24-3721152 


436921 


2388047811 25-7099-303 


8-710983 


595 


354025 


210644875 


24-3926218 


8-410333i 662 


438244 


2901175-281 25-7293607 


8-715373 


596 


355216 


21170873e 


24-4131112 


8-415542! 663 


439559 


2914342471 25-7487854 


8-719760 


597 


356409 


212776173 


24-4335834 


8-42024fr 664 


440396 


2937519441 25768 1975 


8724141 


598 


357604 


! 213347192 


24-4540385 


8-4-24945^ 655 


442225 


294079625! 25-7875939 


8-728518 


599 


353801 


214921799 


24.4744765 


8-4296331 666 


443556 


295408296' 25 8069758 


8-732892 


601 


360000 


216000000 


24 4943974 


8-434337: 667 


444889 


296740963 25-8363431 


8-737260 


601 


361-201 


217081801 


24-5153J13 


8-4390l0!| 663 


446224 


298077632 25-8455960 


8741625 


602 


362404 


218167-208 


24-5356i33 


8-44368311 669 


447561 


299418309 25-8650343 


8-745985 


603 


363609 


1 219256227 


24-5560533 


8 41836UI' 670 


448900 


300763000 25-8843582 


8-750340 



TABLE OF SQUARES, CUBES, AND ROOTS. 



643 



No, Square. 



67 J 
672 
673 

674 
675 
676 

677 

678 

679 

680 

681 

682 

683 

684 

685 

686 

687 

688 

689 

690 

691 

692 

693 

t>94 

695 

696 

697 

698 

699 

700 

701 

702 

703 

704 

705 

706 

707 

708 

709 

710 

711 

712 

713 

714 

715 

716 

717 

718 

719 

720 

721' 

' 7-/31 
724| 
725! 
726 1 
727 
728 
729 
730 
731 
732 
733 
734 
735 
736 
737 



Cube. 



Sq. Root. ICubeRoot. 



450241 

451584 

452929 

454276 

455625 

45697G 

458329 

459684 

461041 

46240U 

463761 

465124 

466489 

467856 

469225 

470596 

471969 

473344 

474721 

476100 

477481 

478864 

480249 

48163C 

483025 

484416 

4858C9 

487201 

488601 

490000 

491401 

492^04 

494209 

495616 

497025 

498436 

499849 

501264 

502681 

504100 

5U5521 

506944 

508369 

5097^6 

511225 



515524 
516961 
518400 
519841 
521284 
522729 
524176 
525625 
527076 
528529 
529984 
5314411 
532900 



302111711 
303464448 
3U4821217 
306182024 
307546875 
308915776 
310288733 
311665752 
313046839 
314432000 
315S21241 
317214568 
31861198- 
320013504 
321419125 
322828856 
3242427U3 
3^5661672 
327082769 
328509000 
3299393' 
331373888 
332812557 
3342553^S4 
335702U75 
337153536 
338608873 
34006839:^ 
341532099 
343000000 
344472101 
345948408 
34742892' 
348913664 
35040-2625 
351895816 
353393243 
334894912 
356400829 
357911000 
359425131 
360944128 
362407097 
363994344 
365525875 
5126561 367061696 
514089 368601813 
37014623:i 
371694959 
373248U00 
374805361 
37636704^ 
377933067 
379503424 
381078125 
382657176 
38424058;: 
385828352 
387420489 
389017C00 
5343611 390617891 
535S24I 392223168 
5"^72H9' 393832837 
53:^756 3y54469t>4 
540225' 397065375 
541 69.)! 398688256 
543169; 400315553 



25-9035677I 8-754691 
25-9229628! 8-75903S 
25-9422435 
25-9615100 



No. Square. 



25-9807621 

26 0000000 

26-0192237 

26-0384331 

26.0576-284 

26-0768096 

26-0959767 

26-1151-297 

•26-134-2687 

26-1533937 

26-1725047 

26-1916017 

26-2106848 

26--2297541 

26-2488095 

26-2678511 

26-2868789 

26-3058929 

26-3248932 

26-3438797 

26-3628527 

26-3818119 

26-4007576 

26-4196896 

26-4386081 

26-4575131 

26-4764046 

26-4952826 

26-5141472 

26 5329983 

26-5518361 

26-5706605 

26-5894716 

26-6082694 

26-6-270539 

26-6458-<>5-i 

26-6645833 

26-6833281 

26-7020598 

26-7207784 

?" -7394839 

26-7581763 

26-7768557 

26-7955-220 

26-8141754 

26-83-2815' 

26-8514432 

26-8700577 

26-8886593 

26-9072481 

26-9258240 

26-9443872 

26-9629375 

26-9814751 

27-0000000 

27-0185122 

27-0370117 

27-0554985 

27-073'j727 

27-0924344 

27.1108834 

27 1293199 

27 1477439 



8-763381; 

8-767719 

8-772053 

8-776383 

8-780708 

8-785030 

8-7893471 

8-793659! 

8-7979681 

8-80-2272 

8-8J6,572| 

8-810868! 

8-815160 

8-819447 

8-823731 

8-8-28010 

8-832285 

8-836556 

8-840823 

8 845085 

8-84i)344| 

8-853598 

8-857849 

8-86-2095 

8.866337 

8-870576 

8-874810 

8-879040 

8-883266 

8-887483 

8-891706 

8-895920 

8-900130 

8-904337 

8-908539 

8-912737 

8-916931 

8-921121 

8-925308 

8-929490 

8-933669 

8-937843 

8-942014 

8-946181 

8-950344 

8-954503 

8-958658 

8-962809 

8-966957 

8-971101 

8-975241 

8-979377 

8-983509 

8-987637 

8-991762 

8-995883 

9-000000 

9-004113 

9-008223 

9-012329 

9-016431 

9-020529 

y 02462 4 

9-028715 

9-032802 



738 

739 

740 

741 

742 

743 

744 

745 

746 

7^17 

748 

749 

750 

751 

752 

753 

754 

755 

756 

757 

758 

759 

760 

761 

762 

763 

764 

765 

766 

767 

768 

769 

770 

771 

772 

773 

774 

775 

776 

777 

778 

7 

780 

781 

782 

783 

784 

785 

786 

78/ 

788 

789 

790 

791 

792 

793 

791 

795 

796 

79 

798 

799 

800 

801 

802 

803 

804 



Cube. 



544644 
546121 
547600 
549081 

550564 

552049 

553536 

555025 

556516 

558009 

559504 

561001 

562500 

564001 

565504 

567009 

568516 

570025 

571536 

573049 

574564 

576081 

57760C 

579121 

580644 

582169 

583696 

585:^25 

586756 

588289 

589824 

591361 

592900 

59444 

595984 

597529 

599076 

600625 

6021761 

603729] 

605-284 

606841 

6084U0 

60J961 

611524 

613089 

614656 

616-225 

617796 

619369 

620944 

622521 

624100 

625681 

627264 

6-28849 

630436 

632025 

633616 

635-209 

636804 

638401 

640000 

641601 

643204 

644809 

646416 



Sq. Rod. 



401947272 

403583419 

405224000 

406869021 

408518488 

4101724071 

411830784 

413493625 

415160936! 

416832723 

418508992 

420189749 

421875000 

423564751 

425259008 

426957777 

428661064 

430368875 

432081216 

433798093 

435519512 

437245479 

438976000 

440711081 

4424507-28 

444194947 

445943744 

447697125 

449455096 

451217663 

452984832 

454756609 

456533000 

458314011 

460099648 

461889917 

4636848-24 

465484375 

467288576 

469097433 

470910952 

47i'7-29139 

47455-2000 

47637y541 

470211768 

480048687 

481890304 

4837366: 

485587656 

487443403 

489303872 

491169069 

4y 30 39000 

494913671 

496793080 

49867725/ 

500566184 

502459875 

504358336 

506-261573 

508169592 

510082399 

512000000 

51392-2401 

515849608 

517781627 

519718464 



CubeRooL 



9-036886 
9040965 
9-045042 
9-049114 
9-053183 



27-1661554 

27-1845544 

27-2029110 

27-2213152 

27-2396769 

27-2580263; 9-057248 

27-27636341 9061310 

27-2946881 

27-3130006 

27-3313007 

27-3495887 

27-3678644 



27-3861279 

27-4043792 

27-42-26184 

27-4408455 

27-4590604 

27-4772633 

27-4954542 

275136330 

27-5317998 

27-5499546 

27-5680975 

27-5862284 

27-6043475 

276224546 

27-640549d 

■27-6586334 

27-6767050 

27-6947648 

27-71-28129 

27-7308492 

27-7488739 

27-7668868 

27-784:5880 

27-8028775 

27-8208555 

27-8388218 

27-8567766 

27-8747197 

27-8926514 

27-9105715 

27-9284801 

27-9463772 

27-9642629 

27-9821372 

28-0000000 

28 017o515 

28-0356^15 

28-0535203 

28-0713377 

28-0891438 

28-l06i)386 

28-1247222 

28-1424946 

28-1602557 

28-1780056 

28-1957444 

28-2134720 

28-2311884 

28-2488933 

28-2665881 

28-2842712 

28-3019434 

28-319tj045 

28-3372546 

28-3548938 



9-065368 

9 069422 

9-073473 

9077520 

9 081563 

9-085603 

9 08963y 

9093672 

9-097701 

9-101726 

9-105748 

9-109767 

9-113782 

9-117793 

9-121801 

9-125805 

9-129806 

9-133803 

9-l377y 

9-141787 

9-145774 

9-149750 

9-15373' 

9-.»577l4 

9-16168 

9-165656 

9-16962:i 

9-17358J 

9-177544 

9-181500 

9-185453 

9-18040. 

9-ly33i7 

9-197290 

9-20r22i, 

9-205i0. 

9-20y09t 

9-2130-25 

9-2l6y5v. 

9-2208/ 

9-z-2-i7Ji 

9-2-2:370< 

y-2020li. 

9-2Jb52b 

9-210435 

9-244333 

9 -248231 

9-25213V. 

9-250022 

9-2:^9011 

9-26.)7y 

9-267680 

9-27 155y 

9-275435 

9-2793UO 

9-283170 

9-237044 

9-290.^07 

9-29476- 

9-298624 



644 



APPENDIX. 



No. 

805 


Square. 1 Cube. | Sq. Root. CubeRoot.' 


No. 


Square. 


Cute. 1 Sq. Root. jCubeRool] 


648025 


521660125 28 3725219 


9-302477 


872 


760334 


0030548481 


29-5296401 


9-553712 


806 


649636 


5236U0616I 28 3901391 


9-300328 


873 7621291 


055338617 


29-5465734 


9-557363 


807 


651249 


5255579431 284077454 


9-310175 


874 


763376 


6676276-24 


29-5634910 


9-561011 


808 


652864 


5275141121 28 4253408 


9-3140191 


875 


765525 


669921875 


29-5803989 


9-564650 


809 


654481 


529475129 


28-4429253 


9-317860 


876 


767370 


()72221376 


29-5972972 


9-568298 


810 


656100 


531441000 


28-4604989 


9-321097 


877 


709129 


674526133 


29-6141858 


9-571938 


811 


657721 


533411731 


28-4780017 


9-3255321 


878 


770884 


670830152 


29-6310648 


9-575574 


812 


659314 


535387328 


28-4956137 


9-32J363 


879 


772641 


079151439 


29-6479342' 9-5792081 


813 


C60969 


537367797 


28-5131549 


9-3331921 


880 


774400 


0814720C0 


29-66479391 


9-58-2840 


814 


662596 


539353144 


28-5300852 


9-337017; 


881 


776161 


083797841 


29-6816442 


9-:86468 


815 


664225 


541343375 


28-5482048 


9-340839 


882 


777924 


080128968 


29.6984843 


9-590094 


816 


665856 


543338496 


28-5657137 


9-3446571 


883 


779689 


688465387 


29-7153159 


9-593717 


817 


667489 


545335513 


28-5832119 


9-3-8473' 


8-Srl 


781456 


690807101 


29-7321375 


9-597337 


818 


669124 


547313432 


28-6006993 


9-352-280 


8-35 


783225 


6931541-25 


29-7489496 


9-600955 


819 


670761 


549353259 


28-6181760 


9-356095 


886 


78499B 


095506456 


29-7657521 


9-604570 


820 


672400 


551368000 


28-6356421 


9-359902 


887 


786769 


697854103 


29-7825452 


9-608182 


821 


674041 


553387661 


28-6530976 


9-3037051 


838 


788544 


700227072 


29-7993289 


9-611791 


822 


675684 


555412248 


28-0705424 


9307505 


889 


790321 


702595369 


29-8161030 


9-61539'' 


823 


677329 


557441767 


28-0879760 


9-371302 


890 


792100 


704969000 


29-8328678 


9-619002 


824 


078976 


559476224 


28-7054002 


9-375090 


891 


793881 


707347971 


29-8496231 


9-622003 


825 


680625 


561515625 


28-72-28132 


9-378887 1 


892 


795664 


709732288 


29-8663690 


9-626202 


826 


682276 


563559976 


28-7402157 


9-382075 


893 


797449 


712121957 


29-8831056 


9-629797 


827 


683J29 


565609283 


28-7570077 


9-380460! 


894 


799236 


714516984 


29-8998328 


9-633391 


828 


685584 


567663552 


28-7749S91 


9-390242 


895 


801025 


716917375 


29-9165506 


9-636981 


829 


687241 


569722789 


28-7923001 


9-39402 li 


896 


802816 


719323136 


29.-33-2591 


9-040569 


830 


6889)0 


571787000 


28-8097206 


9-3..7796 


897 


804609 


721734273 


29-9499533 


9-044154 


831 


690561 


573856191 


28-8270706 


9-4015691 


898 


806404 


724150792 


29-9666481 


9-047737 


832 


692224 


575930368 


28-8444102 


9-4053391 


899 


808201 


72657269S 


29-9S33-.J87 


9-051317 


833 


693889 


578009537 


28-8617394 


9-4091051 


900 


810000 


729000000 


30-0000000 


9-054894 


834 


695555 


580093704 


28-8790582 


9-4128691 


901 


811801 


73143-2701 


30-0166620 


9-658408 


835 


697225 


582182875 


28-8903666 


9-416630 


902 


813604 


733870808 


30-0333148 


9-00-2040 


836 


698896 


584277056 


28-9136646 


9-4203871 


903 


815409 


736314327 


30-0499584 


9-005010 


837 


700569 


586376253 


28-9309523 


9-424142 


904 


817216 


738763264 


30-0665923 


9-669176 


838 


702244 


588480472 


28-9482297 


9-4278d4i 


905 


819025 


741217625 


30-0832179 


9-672740 


839 


703921 


590589719 


28-9654967 


9-431642' 


906 


820836 


743677410 


30-0998333 


9-676302 


840 


705600 


592704000 


28-9827535 


9-435388 


907 


822649 


746142643 


30-1164407 


9-679860 


841 


707281 


594823321 


29-0000000 


9-439131' 


908 


824464 


748613312 


30-1330383 


9-r.83417 


842 


708964 


596947688 


29 0172363 


9-442870 


909 


826-281 


751089429 


30-1496263 


9-686970 


843 


710649 


599077107 


29-0344623 


9-446607 


910 


828100 


753571C00 


30-1662063 


9-690521 


844 


712336 


601211584 


29-0516781 


9-450341 


911 


829921 


756053031 


30-18-27765 


9-694069 


845 


714025 


0U3351125 


29-0688837 


9-454072 


912 


831744 


758550528 


30-1993377 


9-697615 


846 


715716 


005495736 


29-0860791 


9-457800 


913 


833569 


761048497 


30-2158399 


9-701158 


847 


717409 


607645423 


29-1032644 


9-4615-25 


9U 


835396 


763551944 


30-232432^ 


9-704699 


848 


719104 


609800192 


29-1204396 


9-465247 


915 


837225 


766060875 


30-2489669 


9-708237 


849 


720801 


611960049 


29-1376046 


9-468966 


916 


839056 


768575296 


30-2654919 


9-711772 


850 


722500 


614125000 


29-1547595 


9-472682 


917 


840889 


771095213 


30-2820079 


9-715305 


851 


724201 


616295051 


29-1719043 


9-476390 


918 


842724 


773620632 


30-2985148 


9-718835 


852 


725904 


618470208 


29-1890390 


9-480106 


919 


844561 


77615 155S 


30315012a 


9-722363 


853 


727609 


620650477 


29-2061637 


9-483814 


921 


846400 


77863300L 


30-3315018 


9-725388 


854 


729316 


022835804 


29-2232784 


9-487518 


921 


848241 


781229961 


30-3479818 


9-729411 


855 


7310^5 


025026375 


29-2403830 


9-491220 


922 


850084 


783777448 


30-364452<J 


9-73-2931 


856 


73273r 


627222016 


29-2574777 


9-494919 


923 


851929 


786330467 


30 3809151 


9-736448 


857 


734449 


029422793 


29-2745623 


9-498615 


924 


853776 


1 788889024 


30-3J73682 


9-739963 


858 


730164 


631628712 


29-2916370 


9-502308 


925 


855625 


791453125 


30-4138127 


9-743476 


859 


737881 


633839779 


29-3087018 


9-505998 


926 


857476 


79402277( 


30-4302431 


9-746986 


860 


739600 


030050000 


29-3257566 


9-509685 


927 


859329 


796597982 


30-4466747 


9-750493 


861 


741321 


038277381 


29-3428015 


1 9-513370 


928 


861184 


79917875-2 


30-4630924 


9-753398 


862 


743044 


040503928 


29-3598365 


9-517051 


929 


863041 


801765081 


30-4795012 


9-757500 


863 


744769 


042735647 


29-3768010 


i 9-520730 


930 


864900 


8C4357oa 


30-49590141 9-761000 


864 


746496 


644972544 


29-3938709 


1 9-524405 


931 


866761 


i 600954491 


30-5122926! 9-764497 


865 


748225 


047214025 


29-4108823 


9-528079 


932 


8686241 80955756fc 


30-52857501 9-767992 


866 


749956 


049401896 


29-4278779 


9-531750 


933 


87J4H- 


j 812)06237 


30-5450487 


9-771484 


867 


75188y 


051714303 


29-4448037 


1 9-535417 


934 


87235t 


814780504 


30-5514136 


9-774974 


868 


753424 


053972032 


29-4018397 


1 9-539082 


935 


874-225 


, 817400375 


30-5777697 


9-778402 


869 

ti7C 


i 755161 


656234909 


j 29-4788059 


, 9-542744 


936 


87609G 


1 820025y5f 


30-5,41171 


9 731947 


756901 


65850300C 


29-4957024 


! 9-546403 


937 


877909 


1 82265595C 


30-610455? 


9 785429 


87 li 758641 


6607763111 29-51270C1 


1 9-550059 


93a 


879844 


1 82529367-<> 


30-6207857 


1 9-78S909 



TABLE OF SQUARES, CUBES, AND ROOTS. 



645 



No. 

939 


Square. 


Cube. 


Sq. Root. 


CubeRoot. 


No. 


Square. 


Cube. 


Sq. Root. 


C5beRooiJ 


831721 


82793G019 


30-6431069 


9 792386 


970 


940900 


9126730CO 


31-1448230 


9-898983 


910 


8b3600 


830584000 


30-6594194 


9-795361 


971 


942 ■i41 


915498611 


31-160372.* 


9-902333 


941 


8H5481 


833237621 


30-6757233 


9-799334 


972 


944784 


918330048 


311769145 


9-905782 


912 


83"'364 


835S9688S 


30-6920185 


9-802304 


973 


946729 


921167317 


31-1929479 


9-909178 


943 


889249 


838561807 


30-7083051 


9-806271 


974 


948676 


924010424 


31-2039731 


9-912571 


944 


891136 


841232334 


30-7245330 


9-809736 


975 


950625 


926859375 


31-2249900 


9-915962 


945 


893025 


843D08625 


30-7408523 


9-813199 


976 


952576 


929714176 


31-2409987 


9-919351 


946 


894916 


84G590536 


30-7571130 


9-816659 


977 


954529 


932574833 


31-2569992 


9-922738 


947 


8'J6S09 


849278123 


30-7733651 


9-820117 


978 


956484 


935441352 


31-2729915 


9-926122 


948 


898704 


851971392 


30-7896086 


9-823572 


979 


958141; 933313739 


31-288.'757 


9-929504 


949 


900601 


854670349 


30-8053436 


9-827025 


980 


960400 91 1192000 


31-3049517 


9-932834 


950 


90:-500 


857375000 


30-8220700 


9-830476 


931 


96236 l| 944076141 


31-3209195 


9-936261 


951 


904101 


860085351 


30 8382879 


9-8339-24 


982 


964.3241 946966168 


31-3368792 


9-939636 


952 


9U6304 


862801408 


30-8544972 


9-837369 


983 


966289! 949862087 


31-3528308 


9-943009 


953 


908209 


865523177 


30-8706981 


9-840813 


934 


9682561 95-2763-)04 


31-3687743 


9-916380 


954 


910116 


868250664 


30-8868904 


9-844254 


935 


970225 955671625 


31-3847097 


9-949748 


955 


912025 


870983875 


30-9030743 


9-8476:12 


986 


972196 958535256 


31-4006,369 


9-953114 


956 


913936 


873722816 


30-9192497 


9-8511^:8 


987 


974169! 961504803 


31-4165561 


9-956477 


957 


915849 


876467493 


30-9354166 


9-854562 


238 


976144 


9644302^2 


31-43246731 9-959839| 


958 


917764 


879217912 


30-9515751 


9-857993 


989 


978121 


967361669 


31-4483704 


9-963 19S 


959 


919681 


831974079 


30-9677251 


9-861422 


990 


980100 


970299000 


31-4642654 


9-96655.5 


960 


92ia)0 


884736000 


30-9838668 


9-864843 


991 


982081 


973242271 


31-4801525 


9-969909 


961 


923521 


887503G31 


310000000 


9-868272 


992 


984064 


976191488 


31-4960315 


9-973262 


962 


925444 


^•90277128 


310161248 


9-871694 


993 


986049 


979146657 


31-5119025 


9-976612 


963 


927369 


893056347 


310322413 


9-875113 


994 


988036 


982107784 


31-5277655 


9-979960 


964 


9292^.6 


5J5341344 


31-0483494 


9-878530 


995 


990-025 


985071875 


31-5436206 


9-933305 


965 


931225 


898632125 


31-0644491 


9-881945 


996 


992016 


983047936 


31-5594677 


9-986649 


966 


933156 


901423696 


310805405 


9-835357 


997 


994009 


991026973 


31-5753068 


9-989990 


967 


935089 


904231063 


31-0966235 


9-883767: 


993 


996004 


994011992 


31-5911330 


9-993329' 


968 


937U24 


907039232 


31-1126984 


9-892175 


999 


993001 


997002999 


31-6069613 


9-996666, 
10000000 


969 


93S961 


909853209 


31-1287648 


9-895530 


1000 


1000000 ! 1000000000 


31-6227766 



The following rules are for finding the squares, cubes, and roots of num- 
bers exceeding 1000. 

To find the square of any number divisible without a remainder. Rule. — Di- 
vide the given number by such a number from the foregoing table as will 
divide it without a remainder ; then the square of the quotient, multiplied by 
the square of the number found in the table, will give the answer. 

Exatnple. — What is the square of 2000 ? 2000, divided by 1000, a number 
found in the table, gives a quotient of 2, the square of which is 4, and the 
square of 1000 is 1,000,000, therefore : 

4 X 1,000,000 = 4,000,000: the Ans. 

Another Example. — What is the square of 1230? 1230, being divided by 
123, the quotient will be 10, the square of which is 100, and the square of 123 
is 15,129, therefore : 

100 X 15,129 = 1,512,900: the Ans. 

To find the square of any number not divisible without a remainder. Rule. — 
Add together the squares of such two adjoining numbers from the table as 
shall together equal the given number, and multiply the sum by 2 ; then this 
product, less i, will be the answer. 

Example. — What is the square of 1487 ? The adjoining numbers, 743 and 
744, added together, equal the given number, 1487, and the square of 743 = 
552,049, the square of 744 = 553,536, and these added = 1,105,585, therefore : 

1,105,585 X 2 = 2,211,170 — I = 2,211,169 : the Ans. 

To find the cube of any number divisible without a remainder. Rule. — Divide 
the given number by such a number from the foregoing table as will divide 



646 APPENDIX. 

it without a remainder ; then the cube of the quotient, multiplied by the cube 
of the number found in the table, will give the answer. 

Z'jr^w//^.— What is the cube of 2700? 2700, being divided by 900, the quo- 
tient is 3, the cube of which is 27 and the cube of 900 is 729,000,000, there- 
fore : 

27 X 729,000,000 = 19,683,000,000 : the Ans, 

To find the square or cube root of numbers higher than is found in the table, 
/^uk.— Select, in the column of squares or cubes, as the case may require, that 
number which is nearest the given number ; then the answer, when decimals 
are not of importance, will be found directly opposite, in the column of num- 
bers. 

£xamJ>le.—Whsit is the square root of 87,620? In the column of squares, 
87,616 is nearest to the given number ; therefore, 296, immediately opposite 
in the column of numbers, is the answer, nearly. 

Another example. — What is the cube root of 110,591? In the column of 
cubes, 110,592 is found to be nearest to the given number ; therefore, 48, the 
number opposite, is the answer, nearly. 

To find the cube root more accurately. Rule. — Select from the column of 
cubes that number which is nearest the given number, and add twice the 
number so selected to the given number ; also, add twice the given number 
to the number selected from the table. Then, as the former product is to the 
latter, so is the root of the number selected to the root of the number given. 

Example. — What is the cube root of 9200? The nearest number in the col- 
umn of cubes is 9261, the root of which is 21, therefore : 

9261 9200 



18522 18400 

9200 9261 



As 27,722 is to 27,661, so is 21 to 20-953 -f , the Ans. 

Thus, 27,661 X 21 = 580,881, and this divided by 27,722 = 20-953 +. 

To fijid the square or cube root of a whole number with decimals. Rule. — Sub- 
tract the root of the whole number from the root of the next higher number, 
and multiply the remainder by the given decimal ; then the product, added to 
the root of the given whole number, will give the answer correctly to three 
places of decimals in the square root, and to seven in the cube root. 

Example. — What is the square root of 11-14? The square root of 11 if 
3-3166, and the square root of the next higher number, 12, is 3-4641 ; the for- 
mer from the latter, the remainder is 0-1475, and this by 0-14 equals 0-02065. 
This added to 3-3166, the sum, 3-33725, is the square root of 11-14. 

To find the roots of decimals by the use of the table. Rule. — Seek for the 
given decimal in the column of numbers, and opposite in the columns of roots 
will be found the answer, correct as to the figures, but requiring the decimal 
point to be shifted. The transposition of the decimal point is to be performed 
thus : For every place the decimal point is removed in the root, remove it in 
the number two places for the square root and three places for the cube root. 



J 



THE REDUCTION OF DECIMALS. 647 

Examples. — By the table, the square root of 86 -o is 9-2736, consequently by 
the rule the square root of o-86 is 0-92736. The square root of 9* is 3-, hence 
the square root of 0-09 is 0-3. For the square root of 0-0657 we have 
0*25632, found opposite No. 657. So, also, the square root of 0-000927 is 
0-030446, found opposite No. 927. And the square root of 8-73 (whole num- 
ber with decimals) is 2-9546, found opposite No. 873. The cube root of o-8 
is 0-928, found at No. 800 ; the cube root of 0-08 is 0-4308, found opposite 
No. 80, and the cube root of 0-008 is 0-2, as 2-0 is the cube root of 8-0. So 
also the cube root of 0-047 is 0-36088, found opposite No. 47. 



RULES FOR THE REDUCTION OF DECIMALS. 

To reduce a fraction to its equivalent deci?nal. Rule. — Divide the numerator 
by the denominator, annexing cyphers as required. 

Example. — What is the decimal of a foot equivalent to three inches ? 
3 inches is jg- of a foot, therefore : 

^i , . . 12)3.00 

•25 Ans. 
Another example, — What is the equivalent decimal of | of an inch? 

I . . . 8)7-000 

•875 Ans. 

To reduce a compound fraction to its equivalent decijnal. Rule. — In accordance 
with the preceding rule, reduce each fraction, commencing at the lowest, to 
the decimal of the next higher denomination, to which add the numerator of 
the next higher fraction, and reduce the sum to the decimal of the next higher 
denomination, and so proceed to the last ; and the final product will be the 
answer. 

Exajnple. — What is the decimal of a foot equivalent to five inches, f and ^ 
of an inch ? 

The fractions in this case are, \ of an eighth, f of an inch, and -/^ of a foot, 
therefore : 



\ 2) i-o 

•5 
3- eighths. 

f 8) 3 • 5000 

•4375 
5. inches. 

■h 12) 5-4 37500 

-453125 Ans. 

The process may be condensed, thus : write the numerators of the given 



648 APPENDIX. 

fractions, from the least to the greatest, under each other, and place each de- 
nominator to the left of its numerator, thus : 



10 

3 • 5000 
5-437500 



•453125 Ans. 

To reduce a decimal to its equivalent in terms of lower denominations. Rule. 
— Multiply the given decimal by the number of parts in the next less denomi- 
nation, and point off from the product as many figures to the right hand as 
there are in the given decimal ; then multiply the figures pointed off by the 
number of parts in the next lower denomination, and point off as before, and 
so proceed to the end ; then the several figures pointed off to the left will be 
the answer. 

Example, — What is the expression in inches of 0-390625 feet ? 

Feet 0-390625 

12 inches in a foot. 



Inches 4-687500 

8 eighths in an inch. 

Eighths 5-5000 

2 sixteenths in an eighth. 

Sixteenth i-o 

Ans., 4 inches, f and -fg-. 

Another example. — What is the expression, in fractions of an inch of 
0'6875 inches? 

Inches 0-6875 

8 eignths in an inch. 

Eighths 5 • 5000 

2 sixteenths in an eighth. 

Sixteenth 10 

Ans., t and -j^. 



\ 



TABLE OF CIRCLES. 

(From Gregory's Mathematics.) 

From this table may be found by inspection the area or circumference of a 
circle of any diameter, and the side of a square equal to the area of any given 
circle from i to lOO inches, feet, yards, miles, etc. If the given diameter is ia 
inches, the area, circumference, etc., set opposite, will be inches ; if in feet, 
then feet, etc. 









Side of 








Side or 


Diam. 


Area. 


Circum. 


equal sq. 


Diam. 


Area. 


Circum. 


cfjual scj. 


'¥j 


•01908 


•78539 


•22155 


-75 


90-76257 


3377212 


9-5-2693 


•5 


•19635 


1-57079 


-44311 


11- 


95-03317 


34-55751 


9-74349 


•75 


•44178 


1 2-35619 


•66467 


-25 


99-40195 


35-34-291 


9-97005 


1- 


•7853'J 


3-14159 


•886-22 


•5 


103-86890 


36-12S31 


10-19160 


•25 


1-22718 


3-92699 


1-10778 


•75 


108-43403 


38-91371 


10-41316 


•5 


1-7G714 


4-71-238 


1-32934 


12- 


113 09733 


37-69911 


10-63472 


•75 


2-40528 


5-49773 


1 --55983 


-25 


llr85381 


38-43451 


1085627 


2- 


314159 


6-23318 


1-77245 


-5 


1^22-71846 


39-26990 


11-07783 


•25 


3-97fi07 


7-06358 


1-99401 


-75 


127-676'28 


4005530 


11-29939 


•5 


4-90373 


7-85393 


2-21556 


13- 


132-73228 


40-84070 


ll-'>2iYJ5 


•75 


5-93957 


8-63937 


2-43712 


-25 


137-83646 


41-6-2610 


11-74-250 


3^ 


7-0685S 


942477 


2-65388 


-5 


143-13331 


4-2-4 1150 


11-98406 


•25 


8-29576 


10-21017 


2-88023 


-75 


143-48934 


43-19639 


12-18562 


•5 


9-62112 


10-99557 


310179 


14- 


153-93804 


43 98-2-29 


12-40717 


•75 


1104466 


11-78097 


3-3-2335 


•25 


159-48491 


44-76769 


12-82373 


4- 


12-56637 


12-56637 


3-54490 


•5 


165-1»2996 


45-55309 


12-85029 


•25 


14- 186-25 


13-35176 


3-76646 


•75 


170-87318 


46-33349 


13-07184 


•5 


15-90431 


14-13716 


3-98802; 


15^ 


176-71458 


47-12338 


13-29340 


•75 


17-72054 


14-92256 


4-20957! 


•25 


182-65418 


47-90923 


13-51496 


5- 


19-63195 


15-70796 


4-431 13i 


-5 


188-69190 


48-694C)8 


13-73651 


•25 


21-64753 


16-493.36 


4-65269] 


•75 


194-8-2783 


49-48003 


13-95307 


•5 


23-75329 


17-27875 


4-87424' 


15^ 


201-06192 


50-26548 


14-17963 


•75 


25-96722 


18-06415 


5-095»a 


•25 


207-394-20 


51-05;)83 


14-10118 


6^ 


28-27433 


18-84955 


5-31736 


•5 


213-82464 


51 -833 27 


14-62-274 


•25 


30-67961 


19-63495 


5-53891 


•75 


220-.35327 


52 62167 


14-84430 


•5 


33-18307 


20-42035 


5-760471 


17^ 


226-93008 


53-40707 


15-06535 


•75 


35-78470 


21-20575 


5-9321)31 


•25 


233-70504 


54-19-247 


15-28741 


7- 


33-48156 


21-99114 


6-20358 


•5 


240-52818 


54-97737 


15-50897 


■25 


41-28249 


27-77654 


6^4-2514 
6-64670 


•75 


247-44950 


55 76326 


15-73052 


•5 


41-17861 


23-56194 


18- 


254-46900 


58-54368 


15-95-208 


•75 


47-17-297 


24-34734 


6-86325 


•25 


261-58667 


57-33406 


16-17364 


8- 


50-26548 


25-13274 


7-08981 


•5 


268-80252 


53-1 19 46 


1639519 


25 


53-45616 


25-91813 


7-31137 


-75 


276-11654 


53-90486 


16-61875 


•5 


56-74501 


26-70353 


7-53292 


19- 


233-5-2873 


59-69028 


16-83331 


•75 


6013204 


27-48893 


7-75448 


•25 


291-03910 


60-47565 


17-05936 


9- 


63-61725 


28-27433 


7-976J4 


-5 


298-64765 


61-28105 


17-28142 


•25 


67-20063 


29-0597a 


8-19759 


-75 


306-35437 


62-04645 


17-50298 


•5 


70-83218 


29-84513 


8-41915 


20- 


314-15926 


62-83185 


17-72153 


•75 


74-66191 


30-63052 


8-64071 


•25 


322-06233 


63-617-25 


17-91809 


10^ 


78-53981 


31-41592 


8-86226 


-5 


330-06357 


64-40284 


18-16765 


•25 


82-51539 


32-20132 


9-08382 


-75 


333-16-299 


65-18301 


18-389-20 


•5 1 


86-59014 


32-98672 


9-30538! 


21- 


346-36059 


65-97314 


18-61076 



650 






APPENDIX. 












Side of 






1 


Side of 


Diam. 


Area. 


Circam. 


equal sq. 


Diam. 


Area. 


Circum. 


enual sq. 


Tr25 


354-65635 


66-758841 


18 83232 


~38~~ 


1134-114941 


119-380521 


.33 67662 


•5 


363-05030 


67-54424 


19-05387 


•25 


1149-08660 


120-16591 


33-89317 


•75 


371-54241 


68-32964 


19-27543 


•5 


1164^15642 


120-95131 


34-11973 


22- 


38013271 


69- 11503! 


19-49699 


•75 


1179.32442 


121-73671 


34-34129 


•25 


388-82 117 


69-900431 


19-71854 


39^ 


1194-59060 


122-5-2211 


34-56285 


•5 


3J7-6078-2 


70-68583! 


19-94010 


•25 


1209-95495 


1-23-3 J751 


31-78440 


•75 


406-41)263 


71-471231 


20-16166: 


•5 


1225-41748 


124-09-290 


35-00596 


23- 


415-475:'.2 


72-25663 


20-38321 


•75 


1240-97818 


124-87830 


35-22752 


•25 


424-55679 


73-04202 


20-60477 


40^ 


1255-63704 


125-66370 


35 44907 


•5 


433-73613 


73-82742 


20-8-2633 


•25 


1272-39411 


126-44910 


3567063 


•75 


443-01365 


74-61282 


2104788 


•5 


1288-24933 


127-23450 


35-89219 


24- 


452-38934 


75-398-22 


21-26944 


•75 


1304-20273 


128-01999 


36-11374 


•25 


461-8r)320 


76-18362 


21-49100 


41- 


1320-25431 


123-80529 


36-33530 


•5 


471-43524 


76-96902 


2171255 


-25 


1336-40406 


129-59069 


36-55686 


•75 


481-10546 


77-75441 


21-93111 


•5 


1352-65198 


130-37609 


36-77841 


25- 


490-87335 


78-53J81 


22-15567 


•75 


1368-99808 


131-16149 


,36-99997 


•25 


500-74041 


79-32521 


22-377-22 


42- 


1385-44236 


131-94689 


37-22153 


•5 


510-70515 


80-11061 


22-59878 


•25 


1401-98480 


132-73228 


37-44308 


•75 


520-76806 


80 89601 


22-82034 


•5 


1418-62543 


133 51768 


37 66464 


26- 


530-92915 


81-68140 


23-04190 


•75 


1435-36423 


131-30308 


37-88620 


•25 


541-18842 


82-46680 


23-26345 


43^ 


1452-20120 


135-08348 


33-10775 


•5 


551-51586 


83-25-220 


23-48501 


•25 


1469-13635 


135 87338 


33-32931 


•75 


562-00147 


84 03760 


23 70657 


-5 


1486-16967 


135-65928 


38-55087' 


27- 


572-55526 


84-82300 


23-92812 


•75 


1503-30117 


137-44467 


33-77-242 


•25 


583-20722 


85-60839 


24- 14968 


44- 


1520-53084 


133-23007 


38-99398 


■5 


593-1:5736 


86-39379 


24-37124 


•25 


1537-85369 


139-01547 


39-21554 


•75 


604-80567 


87-17919 


24-59279 


•5 


1556-23471 


139-80087 


39-43709 


28- 


615-75216 


87-96459 


24-81135: 


•75 


1572-80890 


140-58527 


3965865 


•25 


626-79582 


8874999 


25-03591 


45- 


1590 43128 


141-37166 


39-88021 


•5 


637-931:65 


89-53539 


25-25745 


•25 


1608-15182 


142-15706 


40-10176 


•75 


649-18066 


90-32078 


25-47i;02 


•5 


1625-97054 


142-94-246 


40-32332 


29^ 


660 51985 


9M0S18 


25-70053 


•75 


164388744 


143-72785 


4054488 


•25 


671-95721 


9 39153 


25-92213 


46- 


1661-90251 


144513-26 


40-76643 


•5 


683-49275 


92-67698 


26-143(59 


•25 


1680-01575 


145-29866 


40-98799 


•75 


6:5-12646 


93-4623S 


26-36525 


•5 


1698-22717 


146-08405 


41-20955 


30^ 


706-85834 


94-24777 


25-58680' 


•75 


1716-53677 


146-86945 


41-43110 


•25 


718 68810 


9503317 


26-80836 


47- 


1734-944M 


147-65485 


41-65266 


•5 


730-61664 


95-81857 


27-02992: 


•25 


1753-45048 


148-44025 


41-87422 


•75 


742 64305 


96-60397 


27-251471 


■5 


1772-05460 


149-2-2565 


42-09577 


31- 


751-76763 
766-99039' 


97-38937 


27-47303 


•75 


1790-75639 


150-01104 


42-31733 


•25 


98-17477 


27-694591 


43- 


1809 55736 


150-79644 


42-53889 


•5 


779-31132 


9896016 


27-91614: 


•25 


1828-45601 


151-53184 


42-76044 


•75 


791-73043 


99-74556 


28-13770; 


•5 


1847-45282 


152-36724 


42-98200 


32^ 


804-24771 


100-53096 


23-35926' 


•75 


1866-54782 


153-15264 


43-20356 


•25 


816-86317 


101-31636 


2.S-58081i 


49- 


1885-74099 


153-93804 


43 42511 


•5 


829-57681 


102-10176 


2S-h0237' 


•25 


1905-83233 


154-72343 


43-64667 


•75 


842-38861 


102-88715 


29i)23J3 


-5 


19-24-42184 


155-50883 


45-86823 


33^ 


855-29859 


10367255 


29-21548 


•75 


1943-90954 


156-29423 


44-08978 


•25 


868-3(;675 


104-45795 


29-46704* 


50- 


1963-49540 


157-07963 


44-31134 


•5 


881-41308 


105-21335 


29-68860' 


•25 


1983-17944 


157-96503 


44-53290 


•75 


894-61759 


106-02875 


29-91015 


•5 


2G02-96I66 


153-65042 


44-75445 


34- 


907-921)27 


106-81415 


30-13171 


•75 


2022-84205 


159-43582 


44-97601 


25 


921-32113 


107-59954 


30-35327 


51- 


2042-82062 


160-221-22 


45-19757 


•5 


934-82016 


108-38494 


30-57482 


•25 


2062-89735 


161-00662 


45-41912 


•75 


943 41736 


109-17034 


30-79638 


•5 


2083-07227 


161-79202 


45-64068 


35^ 


962-11275 


109-95574 


31-01794 


•75 


2103-34536 


162-57741 


45-85-224 


•25 


975-90630 


110-74114 


31-23949 


52^ 


2123-71663 


163-36-281 


46-0g:!80 


•5 


989-79803 


111-52653 


31-46105 


•25 


2144-18607 


164-14821 


46-30535 


•75 


1003-78794 


11231193 


31-68-261 


•5 


2164-75368 


164-93361 


46 5-2691 


36- 


1017-87601 


113 09733 


31-90416 


•75 


2185-41947 


165-71901 


46-74847 


•25 


1032-06227 


11388273 


32-12572 


53^ 


2205-18344 


166-50441 


46-97002 


•5 


1046-34670 


114 66813 


32-34728 


•25 


2227-04557 


167-28980 


47-19158 


•75 


1060-72J30 


115-45353 


32-55883 


•5 


2248-00589 


16807520 


47-41314 


37- 


1075-21008 


1 16-238 J2 


32-7903J 


•75 


2-269-06438 


168-85060 


47-63469 


•25 


1089-78903 


117-02432 


33-01195 


54^ 


2290-22104 


169-61600 


47-85625 


•5 


1104-46616 


117-80972 


33-23350 


•25 


• 2311-475S8 


170-43140 


48-0778'l 


•75 


1119-2414" 


1 18-59572 


33-45506 


•5 


2332-82889 


171-21679 


48-2993(i 



TABLE OF CIRCLES. 



651 









Side of 








Side of 


DiF.m. 


Arcii. 


Circi-.];;. 
17200219 


equal sq. 

48-52092' 


Diam. 


Area. 


Circum, 


e(iuul eq. 
63-36522 


54-75 


"2354 -28008 


~n~ 


4015-1517O 


2-24 -623^7 


55- 


2375-82J44 


172-78759 


48-74248 


•75 


404327883 


225-40J-27 


63-53678 


•25 


2397-47698 


173-57299 


48-96403 


72^ 


4071-50407 


2-26-19467 


63-80833 


•5 


241 9^22209 


174-35839 


49-18559 


•25 


4099-8-2750 


226-98006 


6 1-0-2989 


•75 


2441-06657 


175-14379 


49-40715 


•5 


41-28-24909 


227-7(:546 


64-25145 


56- 


2463 00861 


175-92918 


49-62870 


•75 


4156-76886 


2-2855(:86 


64-47300 


•25 


2485-04887 


176-71458 


49-85026 


73^ 


4185 33681 


229-33626 


64-69456 


•5 


2507-18728 


177-49998 


5007182 


•25 


4214-10-293 


230-12 16i'. 


64-91612 


•75 


2520-4'2337 


178-28538 


50-29337 


•5 


4242-91722 


230-9()7(:6 


65-13767 


57- 


2551-75S63 


179-07078 


50-51493 


•75 


4271-82969 


231-69-245 


<)5-35923 


•25 


2574-19156 


179-85617 


50-73649 


74^ 


4300-84034 


232-47785 


65-58079 


•5 


2596-7-2'267 


180-64157 


50-95304 


•25 


4329-94916 


23326325 


65 80234 


•75 


2619-35196 


181-4-2697 


51-17960 


•5 


4359-15615 


23 4-0481' 5 


66-C2390 


53^ 


264207942 


182 21237 


51-40116 


•75 


4388-46132 


234-83105 


65-24546 


•25 


2664 90505 


182-99777 


51-6-2271 


75^ 


4417-86466 


235-61944 


66-46701 


•5 


2687-8-28S6 


183-78317 


51844-27 


•25 


4447-36618 


236-40484 


66-63857 


•75 


2710-85084 


184-56856 


52-06583 


•5 


4476-96538 


237-19024 


66-91043 


59- 


2733-97100 


185-353J6 


52-28738 


•75 


4506-66374 


237-97564 


67-13168 


■25 


2757-18933 


186-13936 


52-50894 


76^ 


4536-45979 


238-76104 


67-35324 


•5 


2780-50584 


186-92476 


52-73050 


•25 


4566 35400 


239-54613 


67-57180 


•75 


280392053 


187-71016 


52-95-205 


•5 


459634640 


240-33183 


67-79635 


60- 


2327-43338 


ia3-49555 


5317364 


•75 


4626-43696 


241-11723 


63 01791 


•25 


2851-04442 


189-28095 


5339517 


77^ 


465562571 


241-90-263 


68-23947 


•5 


2874-75362 


19;)-06635 


53-61672 


•25 


46S691262 


242-638;)3 


63-46102 


•75 


2898-56100 


190-85175 


53-83328 


•5 


4717-29771 


243-47343 


6868258 


61- 


2922-46656 


191-63715 


54 05984 


•75 


4747-78093 


244-25332 


68-90414 


•25 


2946-47029 


192-42255 


54-28139 


78^ 


477836242 


245 044-22 


69-1-2570 


•5 


2970-572-20 


193-20794 


54-50295 


•25 


4809-04204 


245-82962 


69 34725 


•75 


2994-77223 


193-99334 


54-7-2451 


•5 


4839-81983 


246-61592 


69-56881 


62- 


301907054 


194-77874 


54 946C6 


•75 


4S7()-79579 


247-40042 


69-79037 


•25 


3043 46697 


195-56414 


5516762 


79^ 


4901-66993 


248-18531 


70-01192 


•5 


3067-96157 


196-34954 


55-33918 


•25 


4932-74225 


248-97121 


70-23348 


•75 


3092-55435 


197-13493 


5561073 


•5 


4963-91274 


249-73661 


70-45504 


63- 


3117-24531 


197-9-2033 


55-83229 


•75 


4995-18140 


250-34-201 


70-67659 


•25 


314203444 


198-70573 


56053^5 


80- 


5026-548-24 


251-32741 


70-89815 


•5 


3166-92174 


199-49113 


56-27540 


•25 


5058-01325 


252-11231 


71-11971 


•75 


3 191 •907-22 


200-27653 


56-496 J6i 


•5 


5089-57644 


25289820 


71-34126 


64- 


3216-99087 


201-06192 


56-71852 


•75 


5121-23781 


253-63360 


71-56282 


•25 


3242- 17-270 


201-84732 


56-94007 


81^ 


5152-99735 


254-46900 


71-78433 


•5 


3267-45270 


20263272 


57-16163 


•25 


5184-8551)6 


255-25440 


72-00593 


•75 


3292 83088 


203-41812 


57-333191 


•5 


5216-81095 


256-03J80 


72-22749 


65- 


3318-30724 


204-20352 


5760475 


•75 


5248-86501 


255-82579 


72-44905 


•25 


3313-88176 


204-98392 


57-82630 


82- 


5281-01725 


257-61059 


72-67060 


•5 


3369-55447 


205-77431 


53-04786 


•25 


531326766 


258-39599 


72-89216 


•75 


339532534 


206-55971 


53-26942 


•5 


5345-616-24 


259-18139 


73 11372 


66- 


3421 -19439 


207-34511 


53-49097 


•75 


5378-06301 


259-96679 


73-33527 


•25 


344716162 


203-13051 


53-71-253 


83^ 


5410-60794 


260-75219 


73-55633 


•5 


347322702 


208-91591 


58-93409 


•25 


544325105 


261-53753 


7377839 


•75 


349939060 


209-70130 


59-15584 


•5 


5475-99234 


262-32-298 


73-99994 


67- 


3525-65235 


21048670 


59-37720 


•75 


5508-83180 


263-10333 


74-22150 


25 


3552^01228 


211-27210 


59-59876 


84- 


5541-76944 


263-89378 


74-44306 


•5 


3578-47033 


212-05750 


59-82031 


•25 


5574-80525 


264-67918 


74-66461 


•75 


350502665 


212-84-290 


6004187 


•5 


5607-93923 


265-46457 


74-88617 


68- 


3631-68110 


213-62^30 


60-26343 


•75 


5641-J7139 


266-24997 


75-10773 


•25 


3658-43373 


214-41369 


60-434981 


85^ 


5674-50173 


267-03537 


75-32928 


•5 


3685-28453 


215-19909 


60-70654 


•25 


5707-93023 


267-82077 


75-55084 


•75 


3712 23350 


215-98449 


60-92310 


•5 


5741-45692 


268-60617 


75 77240 


69- 


3-'39-28065 


21676939 


61-14965 


•75 


5775-08178 


269-39157 


75-99395 


•25 


3766-4-2597 


217-555-29 


61-37121 


86^ 


5808-80481 


270-17696 


76 21551 


•5 


379CS-66947 


218-34068 


61-59-277 


•25 


5342-62602 


270-96236 


76-43707 


•75 


332101115 


2il'- 12608 


61-81432 


•5 


5876-54540 


271-74776 


76-65362 


70 


3348-45100 


219-91143 


62-03533 


•75 


5;) 10-56296 


272 53316 


76-88/J18 


•25 


3875-98902 


2-20-69683 


62-25744 


87^ 


5944 67869 


273-31856 


77-10174 


•5 


3'J03-62522 


221-48-228 


62-47899 


•25 


5973-89260 


274-103J5 


77-32329 


•75 


3931-35959 


222-2JV/63 


62-70055 


•5 


6013-20468 


274-88935 


77-54485 


71 


3959-19214 


223-05307 


62-9-2211 


•75 


604761494 


275-67475 


777664J 


•2'- 


3987-12286 


223 83347 


6314366 


88^ 


6082-12337 


276-46015 


77 987 96 



652 



APPENDIX. 









"Side of" 








Side ot 


Diam. 


Area. 


Circum. 


equal eq. 


Diam. 


Area. 


Circum. 


equa) sq. 
83-52688 


l8~25 


6116-72993 


277-24555 


78-20952 


~94-25 


6976-74097 


296-09510 


•5 


6151-43476 


278-03094 


78-43103 


•5 


7012-80194 


296-88050 


83-74844 


•75 


6186-23772 


278-81634 


7865263 


-75 


7050-96109 


297-66590 


83-97000 


89- 


6221 -13885 


279-60174 


78-87419 


95- 


7083-21842 


298-45130 


84-19155 


•35 


6256-13315 


230-33714 


79 09575 


•25 


7125-57992 


299-23670 


84-41313 


•5 


6291-23563 


231-17254 


79-31730 


-5 


7163-02759 


300-02209 


84-63467 


•75 


6326-43129 


281-95794 


79-53836 


•75 


7200-57944 


300-80749 


84-856-22 


90^ 


6361-7-2512 


282-74333 


79-76042 


96- 


7238-22947 


3U 1-59239 


85-0VV78 


•25 


639711712 


233-52873 


79-98193 


•25 


7275-97767 


302-37829 


85-29934 


•5 


6432-60730 


284-31413 


80-20353 


-5 


7313-82404 


303-16369 


85-52089 


•75 


6463-19556 


285-09953 


80-42509 


-75 


7351-76859 


303-94908 


85-74-245 


91^ 


6503-83219 


285-83493 


80-646-59 


97- 


7389-81131 


304-73448 


85-96401 


.25 


653J-65689 


286-67032 


80-85820 


-25 


7427-95221 


305-5 1983 


86-18556 


•5 


6575-51977 


237-45572 


81-03976 


-5 


7466-191-29 


306-305-23 


86-40712 


•75 


6811-53382 


288-24112 


81-31132 


-75 


7504-52853 


307-09068 


86-6-2868 


92^ 


6347-61005 


239-02652 


81-53237 


98- 


7542-96396 


307-87603 


86-85023 


•25 


6633-73745 


289-81192 


81-75443 


-25 


7581-49755 


308-66147 


87-07179 


•5 


67-20-06303 


290-59732 


81-97599 


-5 


76-20-12933 


309-44687 


87-29335 


•75 


6756-43678 


291-33271 


82-19734 


-75 


7653-85927 


310-23227 


87-51490 


93- 


6792-90871 


292-16811 


82-41910 


99- 


7697-68739 


311-01767 


87-73646 


•25 


6829-47831 


292-95351 


8-2-0 1066 


-25 


7736-61369 


311-80307 


87-95802 


•5 


6866-14709 


293-73391 


82-86221 


-5 


7775-63816 


312-58346 


88-17957 


•75 


6902-91354 


294-52431 


83-08377 


•75 


7814-76031 


313-37336 


88-40113 


94- 


6939-77817 


295-30970 


83 3053311100- 


7353-98163 


314-15926 


88-62269 



The following rules are for extending the use of the above table. 

To find the area, circumference, or side of equal square, of a circle having a 
diameter of more than 100 inches, feet, etc. Rule. — Divide the given diameter by 
a number that will give a quotient equal to some one of the diameters in the 
table ; then the circumference or side of equal square, opposite that diameter, 
multiplied by that divisor, or the area opposite that diameter, multiplied by 
the square of- the aforesaid divisor, will give the answer. 

Example. — What is the circumference of a circle whose diameter is 228 
feet ? 228, divided by 3, gives 76, a diameter of the table, the circumference 
of which is 238-761, therefore : 

238-761 
3 



716-283 feet. Ans. 

Another example. — What is the area of a circle having a diameter of 150 
inches? 150, divided by 10, gives 15, one of the diameters in the table, the 
area of which is 176-71458, therefore: 



176.71458 



10 X 10 



17,671-45800 inches. Ans. 



To find the area, circumference, or side of equal square, of a circle having an 
intermediate diameter to those in the table. -/?z^/^.— Multiply the given diameter 
by a number that will give a product equal to some one of the diameters ia 
the table ; then the circumference or side of equal square opposite that diame- 
ter, divided by that multiplier, or the area opposite that diameter divided by 
the square of the aforesaid multiplier, will give the answer. 



CAPACITY OF WELLS, CISTERNS, ETC. 



653 



Example. — What is the circumference of a circle whose diameter is 6|, or 
6-125 inches ? 6-125, multiplied by 2, gives 12-25, one of the diameters of the 
table, whose circumference is 38-484, therefore : 

2 )38-484 
19-242 inches. Ans. 

Another exarnple.—Wh^i is the area of a circle, the diameter of which is 3-2 
feet? 3-2, multiplied by 5, gives 16. and the area of 16 is 201 'OGig, therefore ;- 

5 X 5 =25)201-0619(8-0424+ feet. Ans. 
200 

106 
100 

61 

50 

119 
100 

Note. — The diameter of a circle, multiplied by 3-14159, will give its cir- 
cumference,; the square of the diameter, multiplied by -78539, will give its 
area; and the diameter, multiplied by -88622, will give the side of a square 
equal to the area of the circle. 



TABLE SHOWING THE CAPACITY OF WELLS, CISTERNS, ETC. 

The gallon of the State of New York, by an act passed April 11, 1851, is leqiiired to conform to 
the standard gallon of the United States government. This standard gallon contains 231 cubic 
inches. In conformity with this standard the following table has been computed. 

One foot in depth of a cistern of 

3 feet diameter will contain 52-872 gallons. 



3^^ 

4 

4i 

5 

5l 

6 

6i 

7 

8 

9 
10 



- 71-965 
■ 93-995 
.iiS-963 
,146-868 
,177-710 
211 -490 
2^8-207 
287-861 
375-982 
.475-852 
■587-472 
•845-959 



Note. — To reduce cubic feet to gallons, multiply by 7-48. The weight of a 
gallon of water is 8-355 lbs. To find the contents of a round cistern, multi- 
ply the square of the diameter by the height, both in feet, and this product by 

5-875- 



654 



APPENDIX. 



TABLE OF WEIGHTS. 



MATERIALS USED IN THE CONSTRUCTION OR LOADING OF 

BUILDINGS. 

Weights per Cubic Foot. 

As per Barlow, Gallier, Jlaswell, Hurst, Rankine, Tredgold, Wood 
and the Author. 



Material. 



WOODS. 



Acacia 

Alder 

Apple-tree 

Ash 

Beech 

Birch.... 

Box.. 

" French 

Brazil-wood ' .. 

Cedar ' 27 

" Canadian 47 

" Palestine ' 30 

" Virginia Red I .. 

Cherry j 32 

Chestnut, Horse . ' 29 

" Sweet i 27 

Cork ' .. 

Cypress..... j 27 

'' Spanish ! . . 

Deal, Christiania j . . 

'* English I . . 

" (Norway Spruce). ' 21 

Dogwood t .. 

Ebony ' C9 

Elder 

Elm 

Fir (Norway Spruce) 

" (Red Pine) 30 

" Riga 

Gum, Blue 

" Water 

Hackmatack . 

Hemlock 

Hickory 

Lance-wood 

Larch 

" Red 

/' White ■. 

Lignum-vitse 41 

Locust 4) 

Logwood 

Mahogany, Honduras 35 





« 1 




1 





<; 


H 






> 




< 


51 


46 


51 


38 


51 


60 


57 


49 


ss 


4G 


49 


43 


6^ 


62 




83 




(i4 


35 


31 


57 


52 


^3 


34 




40 


46 


39 


41 


'A^ 


55 


41 




15 


41 


34 




40 




44 




29 


33 


27 




47 


8^ 


7G 




43 


59 


46 


33 


27 


44 


37 




47 




63 




62 




37 


3^ 


26 


58 


49 


6^ 


62 


35 


33 


54 


43 




23 


83 


62 


51 


46 




57 


40 


38 



LIaterial. 



Mahogany, St. Domingo. 

Maple 

Mulberry 

Oak, Adriatic 

" Black Bog 

" Canadian 

" Dantzic 

" English 

" Live 

" Red 

" White 

Olive 

Orange 

Pear-tree 

Pine, Georgia (pitch) 

" Mar Forest 

" Memel and Riga. .. 

" Red 

" Scotch 

" White 

" Yellow 

Plum 

Poplar 

Quince 

Redwood 

Rosewood 

Sassafras 

Satin wood 

Spruce 

Sycamore 

Teak 

Tulip-tree 

Vine 

Walnut, Black 

"• White 

Whitewood 

Yew 



METALS. 

Bismuth, Cast 

Brass, Cast 

'* (Gun-metal) 
" Plate 

Bronze 



487 



534 
524 



55 
41 

45 
62 
63 
54 
47 
54 
68 
51 
50 
58 
44 
42 
48 
43 
32 
37 
39 
28 
33 
45 
30 
44 
23 
45 
30 
5 7 
30 
38 
51 
30 
80 
33 
49 
27 
50 



614 
506 
544 
531 
516 



WEIGHT OF MATERIALS. 



655 



TABLE OF WEIGHTS.— (Conizmted.) 

MATERIALS USED IN THE CONSTRUCTION OR LOADING OF 

BUILDINGS. 

Weights per Cubic Foot. 

^s per Barlow, Gallier, Ilaszucll, Ilurst, Rankiiic, TredgolJ, Wood 
and the Attlhor. 



Material. 












> 
< 


Copper, Cast 


537 


549 

487 
474 

486 

... 

492 
468 
449 

180 

103 
304 
187 
156 
^34 

iig 


543 
550 
644 1 

1206 

1108 
509 
481 
454! 
4751 
480 
709! 
7171 
713 
85l! 
849! 
837j 
488 
453 
975 
1345 
1379 
142 
636 
655 
658 
644 
489 
462 
439 

173 
156 
80i 
277 
171 
139 
129 
160 
102 
138 
107 
100 
105 


'' Hammered 

" Plate 


Gold 




'^ Standard 




Gun-metal . 

Iron, Bar 


475 
434 


"• Cast 


" Malleable 


" Wrought 


474 


Lead Cast 


*' En-Ush Cast 

" LliUed . . 








" " 60" 




" 212° 

Nickel Cast 




Pewter 








" Pure . 




" Rolled 




Silver, Parisian Standard... 




" " " Hammered 




Steel 


486 
4i>6 
429 

165 


Tin, Ca<^t 


Zinc, Cabt 


STONES, EARTHS, Etc. 
Alabaster ... 






57 
250 

155 
122 
124 

'85 


Parytes, Sulphate of 

Basalt .. 


Bath Stone 


Beton Coii:;net 


Blue Stone, Common 

Brick 


"- Fire- 


" N. R. common hard. . . 

" " Salmon 

" Philadelphia Front 





Material. 



Brick-work 

"■ dry 

" in Cement 

" in Mortar 

Caen Stone 

Cement, Portland 

" Roman, Cast 

" " and Sand 

equal parts. . 

Chalk 

Clay... 

" with Gravel 

Coal, Anthracite 

"• Bituminous 

" Cannel 

" Cumberland 

Coke 

Concrete, Cement 

Coquina 

Earth, Common 

" Loamy 

" with Gravel 

Emer}"- 

Feld^^par 

Flagging, Silver Gray 

Flinc. ... 

Glass, Crown 

" Flint 

" Green 

"■ Plate 

" _ White 

Granite 

"■ Aberdeen 

" Egyptian Red 

" Guernsey 

" Quincy 

Gravel 

Grindstone 

Gypsum 

Lime, Unslaked 

Limestone 

" Aubigne 

" Limerick 

Marble 

'• Brocatcl 

" Cw^rrara 



96 



95 



104 
100 
112 
110 
130 
81 
100 

113 
145 
122 
160 
96 
83 
79 
85 
54 
130 
106 
110 
126 
126 
250 
160 
185 
163 
160 
183 
105 
163 
174 
165 
164 
166 
185 
166 
105 
134 
140 
52 
169 
146 
162 
170 
166 
170 



656 



APPENDIX. 



TABLE OF WEIGHTS.— (Confmued,) 

MATERIALS USED IN THE CONSTRUCTION OR LOADING OF 

BUILDINGS. 

Weights per Cubic Foot. 

As per Barlow, Gallier, Jlaszaeli, Hiirst, Rankine, Ti'edgold, Wood 
and the Author. 



Material. 



167 



Marble, Eastchester. . . 

Egyptian 

" French 

" Italian 

Marl 

Masonry 

Mica 

Millstone 

Mortar 

*' dry 

" new 

'* Hair, incl. Lath and 

Nails, per foot sup.{ 7 

" Hair, dry 

*' new 

" Sand 3 and Lime paste 2 

.. u 2 " » " 2 

well beat together. . 

Peat, Hard 

Petrified Wood 

Pitch 

Plaster, Cast 

Porphyry, Green 

Red 

Portland Stone . 

Pumice-stone 

Puzzolana 

Quartz, Crystallized 

Rotten-stone 

Sand, Coarse 

*' Common 

" Dry 

" Moist 

" Mortar 

•• Pit 

*■ Quartz 

"• with Gravel 

Sandstone . . 

Amherst, O 

Belleville, N.J... 

Berea, O 

" Dorchester, N. S. 

" Little Falls, N. J. 

" Marietta, O 

" Middletown, Ct. . 



92 





«• 1 










< 


H 


u 








<; 


178 


1-73 




167 




16« 


i6q 


167 


170 


140 


140 


125 




175 




155 


109 


98 


118 


103 




107 


11 







86 




105 




100 




118 




83 




146 




72 




80 




180 




175 


161 


147 




56 




165 




165 




124 




112 


118 


105 


120 


105 


128 


123 




105 


101 


»7 




172 




126 


1^8 


144 




133 




142 




134 




141 


... 


134 




162 




150 



Material. 



Serpentine.. 

Chester, Pa. 

" Green 

Shingle 

Slate 

'" Common 

" Cornwall 

" Welsh. 

Stone, Artificial 

*' Pavmg 

Stone-work 

Hewn 

Rubble 

Sulphur, Melted 

Tiles, Common plain.. 

Trap Rock 

Tufa, Roman 



MISCELLANEOUS. 



Ashes. Wood 

Bark, Peruvian 

Butter 

Camphor 

Charcoal • 

Cotton, baled 

Fat 

Gunpowder 

Gutta-percha 

Hay. baled 

India Rubber 

Isinglass 

Ivory. 

Plaster of Paris 

Plumbago 

Red Lead 

Resin 

Rock Crystal 

Salt 

Saltpetre 

Snow 

Sugar 

Water, Rain .. 

'' Sea 

Whalebone 



60 



165 
144 
152 

95 
159 
167 
157 
180 
135 
151 
140 
160 
140 
124 
115 
170 

76 



58 

49 

59 

62 

26 

20 

58 

57 

61 

17 

61 

69 

114 

73 

131 

559 

68 

171 

133 

131 

14 

80 

62^ 

64 

81 



INDEX. 



PAGE 

Abscissas of Axes, Ellipse 484 

Abutments, Bridges, Strength of. 227 
Abutments, Houses, Strength of.. 53 

Acute Angle Defined 349, 544 

Acute-angled Triangle Defined... 545 

Acute or Lancet Arch 51 

Algebra, Addition 398 

Algebra, Application of 393 

Algebra, Binomial, Multiplica- 
tion of 409 

Algebra, Binomial, Square of a... 429 
Algebra, Binomial, Squaring a. . . 410 

Algebra Defined 392 

Algebra, Denominator, Least 

Common 404 

Algebra, Division, the Quotient.. 419 
Algebra, Division, Reduction. . . . 419 
Algebra, Division, Reverse of 

Multiplication 418 

Algebra, Factors, Multiplication 

of Two and Three 409 

Algebra, Factors, P*Iultiplication 

of Three 408 

Algebra, Factors, Squaring Differ- 
ence of Two 412 

Algebra, Fractions Added and 

Subtracted 403 

Algebra, Fractions, Denominators 407 
Algebra, Fractions Subtracted... 405 
Algebra, Hypothenuse, Equality 

of Squares on 416 

Algebra, Letters, Customary Uses 

of 396 

Algebra, Logarithms Explained.. 425 
Algebra, Logarithms, Examples in 426 
Algebra, Multiplication, Graphical 408 



PAGE 

Algebra, Progression, Arithmeti- 
cal 432 

Algebra, Progression, Geometrical 435 
Algebra, Proportion Essential. . . . 347 
Algebra, Proportionals, Lever 

Formula , .... 421 

Algebra, Quantities, Addition and 

Subtraction 424 

Algebra, Quantities, Division of. 424 
Algebra, Quantities, Multiplica- 
tion of 424 

Algebra, Quantities with Negative 

Exponents 423 

Algebra, Quantity, Raising to any 

Power 423 

Algebra, Radicals, Extraction of., 425 

Algebra, Rules are General 394 

Algebra, Rules, Useful Construc- 
ting 391 

Algebra, Signs 397 

Algebra, Signs, Arithmetical Pro- 
cess by. 396 

Algebra, Signs, Changed when 

Subtracted 400 

Algebra, Signs, Multiplication of 

Plus and Minus 415 

Algebra, Squares on Right-Angled 

Triangle 417 

Algebra, Subtraction 39S 

Algebra, Sum and Difference, Pro- 
duct of 413 

Algebra, Symbols Chosen at Pleas- 
ure 395 

Algebra, Symbol, Transferring a.. 399 
Algebra, Triangle, Squares on 
Right-angled 417 



558 



INDEX. 



PAGE 

Alhambra, or Red House, Ancient 

Palace of the ii 

Ancient Cities, Historical Ac- 
counts of 6 

Ancient Monuments, their Archi- 
tects 6 

Angle at Circumference of Circle. 358 

Angle Defined 544 

Angle to Bracket of Cornice, To 

Obtain 343 

Angle, To Measure a. Geometry.. 348 
Angle-rib to Polygonal Dome. . . . 223 
Angle-rib, Shape of Polygonal 

Domes 223 

Amulet or Fillet, Classic Mould- 
ing 323 

Antfe Cap, Modern Moulding. .. . 334 

Antique Columns, Forms of 48 

Antiquity of Building 5 

Arabian and Moorish Styles, An- 
tiquities of II 

Arseostyle, Intercolumniations. . . 20 

Arc of Circle Defined 547 

Arc of Circle, Length, Rule for. . 475 

Arc, Radius of. To Find 561 

Arc, Versed Sine, To Find (Geom- 
etry) 561 

Arcade 52 

Arcade of Arches, Resistance in.. 52 
Arcade in Bridges, Strength of 

Piers 52 

Arch 50 

Arch, Acute or Lancet 51 

Arch, Archivolt in 52 

Arch, Bridge, Pressure on 51 

Arch, Building, Manner of 50 

Arch, Catenary 51 

Arch, Construction of 50 

Arch, Definitions and Principles of 52 

Arch, Extrados of 52 

Arch, Form of 50 

Arch, Formation in Bridges 51 

Arch, Hooke's Theory of an-. 50 

Arch, Horseshoe or Moorish 51 

Arch, Impost in. 52 

Arch, Intrados of 52 

Arch, Keystone, Position of 50 

Arch, Lateral Thrust in 52 



PAGE 

Arch, Ogee 51 

Arch, Rampant 51 

Arch, Span of an 52 

Arch, Spring in an 52 

Arch, Stone Bridges 230 

Arch-stones, Bridges, Jointing. . . 233 

Arch, Strength of 50 

Arch of Titus, Composite Order. . 28 

Arch, Uses of 50 

Arch, Voussoir in 52 

Architect and Builder, Construc- 
tion Necessary to 56 

Architect, Derivation of the Word 5 

Architects of Italy, 14th Century. 12 
Architecture, Classic Mouldings 

in 323 

Architecture, Ecclesiastical, Origin 

of 14 

Architecture, Egyptian, Character 

of 33 

Architecture, Egyptian, Features 

of 30 

Architecture, English, Ccttage 

Style 35 

Architecture, English, Early 11 

Architecture, Grecian and Roman 8 

Architecture, Grecian, Historj- of. 6 
Architecture, Hindoo, Character 

of 30 

Architecture, Order, Three Princi- 
pal parts of 14 

Architecture, Principles of 44 

Architecture, Roman, Ruins of. . . 11 

Architecture in Rome Defined. . . 7 

Architecture, Result of Necessity. 13 

Architrave Defined 15 

Area of Circle, To Find 475 

Area of Post, Rule for Finding. . , 90 

Area of Round Post, Rule 90 

Area of Surface, Sliding Rupture, 

Rule 88 

Arithmetical Progression (Alge- 
bra) 432 

Astragal, or Bead, Classic Mould- 
ing 323 

Athens, Parthenon, Columns of. . 48 
Attic, a Small Order, Top of 

Building 15 



INDEX. 



659 



PAGE 

Attic Story, Upper Story 15 

Axes of Ellipse (Geometry) 585 

Axiom Defined (Geometry) 348 

Axis Defined 548 

Balusters, Handrailing, Winding 

Stairs 310 

Baluster, Platform Stairs, Position 

of 250 

Baluster in Round Rail, Winding 

Stairs 313 

Base, Shaft, and Capital Defined. 14 
Bathing, Necessary Arrangements 

for 45 

Baths of Diocletian, Splendor of.. 27 
Bead, or Astragal, Classic Mould- 
ings 323 

Beams, Bearings of, Rules for 

Pressure 75 

Beams, Breaking Weight on 74 

Beams, Framed, Rules for Thick- 
ness 130 

Beams, Framed, Position of Mor- 
tise 236 

Beams, Headers Defined 130 

Beams, Horizontal Thrust, Rules 

for 72 

Beams, Inclined, Effect of Weight 

on 72 

Beams, Load on. Effect of 74 

Beams, Splicing 235 

Beams, Tail, Defined 130 

Beams, Trimmers or Carriage, De- 
fined 130 

Beams, Weight on. Proportion of. 130 
Beams, White Pine. Table of 

Weights 177 

Beams, Wooden, Use of Limited. 154 

Bearings for Girders 141 

Binomials, Multiplication of (Al- 
gebra) 409 

Binomials, Square of (Algebra). . . 429 
Binomials, Squaring (Algebra)... 410 

Bisect an Angle (Geometry) 554 

Bisect a Line (Geometry) 549 

Blocking out Rail, Winding Stairs 301 
Blondel's Method, Rise and Tread 
in Stairs 242 



rAc.ii: 
Bottom Rail for Doors, Rule for 

Width 316 

Bow, Mr., On Economics and 

Construction 166 

Bowstring Girder, Cast -Iron, 

Should not be Used 163 

Brace, Length of, To Find (Geom- 
etry) 579 

Braces, Rafters, etc.. To Find 

Length 5 So 

Braces in Roof, Rule for. Same as 

Rafter 208 

Breaking Weight Defined 84 

Brick or Stone Buildings 37 

Brick Walls, Modern 49 

Bridge Abutments, Strength of.. . 227 

Bridge Arches, Formation of 51 

Bridge Arch-stones, Joints of 233 

Bridges, Construction of Various. 223 
Bridge, London, Age of Piles 

under 229 

Bridge Piers, Construction and 

Sizes 228 

Bridge, Rib-built 224 

Bridge, Rib, Construction of. . . . 225 
Bridge, Rib, Framed, Construction 

and Distance 226 

Bridge, Rib, Radials of 226 

Bridge, Rib, Table of Least Rise 

in 224 

Bridge, Rib, Rule for Area of 225 

Bridge, Rib, Rule for Depth of. . . 226 

Bridge, Roadway, Width of 227 

Bridge, Stone, Arch Construction 230 
Bridge, Stone, Arch-stones, Table 

of Pressures on 230 

Bridge, Stone, Arch, Centres for. 

Bad Construction 229 

Bridge, Arch, Spring of 247 

Bridge, Stone, Strength of Truss- 
ing 232 

Bridge, Weight, Greatest on 225 

Bridge, without Tie-Beam 224 

Bridging, Cross-, Additional 

Strength by 137 

Bridging, Cross-, Defined 137 

Bridging, Cross-, Resistance by 
Adjoining Beams. 139 



66o 



INDEX. 



PAGE 

Building, Antiquity of 5 

Building, Elementary Parts of a.. 46 

Building, Expression in a 35 

Building by the Greeks 35 

Building, Modes of, Defined 9 

Building by the Romans 26 

Building, Style of. Selected 10 Suit 

Destination 35 

Butt-joint on Handrail to Stairs. . 303 
Butt-joint, Handrail, Stairs, Posi- 
tion of 307 

Byzantine St3'le, Lombard 10 

Campanile, or Leaning Tower, 

Twelfth Century 12 

Capital, Uppermost Part of a 

Column 15 

Carriage Beam, Well-Hole in Mid- 
dle, Find Breadth 136 

Carriage Beam, One Header, Rule 

for Breadth 133 

Carriage Beam or Trimmer De- 
fined r30 

Carriage Beam, Rule for Breadth. 132 
Carriage Beam, Two Sets of Tail 

Beams, Rule for Breadth 134 

Caryatides, Description and Ori- 
gin of 26 

Cast-iron Bowstring Girder, 

Should not be Used 163 

Cast-Iron Girder, Load at Middle, 

Size of Flanges. .... 162 

Cast-Iron Girder, Load Uniform, 

Size of Flanges 163 

Cast-iron Girder, Manner of Mak- 
ing a 161 

Cast-Iron Girder, Proper Form... 161 

Cast-iron, Tensile Strength of . . . . 161 

Cast-Iron Untrustworthy 161 

Catenary Arch, Hooke's Theory of. 51 

Cathedral of Cologne ii 

Cathedrals, Domes of 53 

Cathedrals of Pisa, Erection in 

1016 12 

Cavern, The Original Place of 

Shelter , 13 

Cavetto or Cove, Classic Moulding 323 

Cavetto, Grecian Moulding 327 



PAGE 

Cavetto, Roman Moulding 329 

Ceiling, Cracking, How to Pre- 
vent 125 

Centre of Circles, To Find (Ge- 
ometry) 556 

Centre of Gravity, Position of ... . 71 
Centre of Gravity, Rule for Find- 
ing, Examples 71 

Chimneys, How Arranged 42 

Chinese Structure, The Tent the 

Model of 14 

Chord of Circle Defined 547 

Chords Giving Equal Rectangles. 363 

Circle, Arc, Rule for Length of.. . 475 
Circle, Area, Circumference, etc.. 

Examples 652 

Circle, Area, Rule for. Length of 

Arc Given 478 

Circle, Area, To Find 475 

Circle, Circumference, To Find . . 473 

Circle Defined 546 

Circle, Describe within Triangle.. 566 
Circle, Diameter and Circumfer- 
ence 472 

Circle, Diameter and Perpendicu- 
lar 468 

Circle Equal Given Circles, To 

Make 580 

Circle, Ordinates, Rule for 471 

Circle, Radius from Chord and 

Versed Sine 469 

Circle, Sector, Area of 476 

Circle, Segment, Area of. 477 

Circle, Segment from Ordinates. . 470 

Circle, Segment, Rule for Area of. 479 

Circles, Table of 649-652 

Circle through Given Points 559 

Circular Headed Doors 320 

Circular Headed Doors, To Form 

Soffit 321 

Circular Headed Windows 320 

Circular Headed Windows, To 

Form Soffit 321 

Circular Stairs, Face Mould for (i). 282 

Circular Stairs, Face Mould for (2). 285 

Circular Stairs, Face Mould for (3). 287 
Circular Stairs, Face Mould, First 

Section ... 283 



INDEX. 



66 1 



Circular Stairs, Falling Mould for 

Rail 2S1 

Circular Stairs, Handrailing for . . 278 

Circular Stairs, Plan of 279 

Circular Stairs, Plumb Bevel De- 
fined 282 

Circular Stairs, Timbers Put in 

after Erection 253 

Cisterns, Wells, etc., Table of 

Capacity of 653 

City Houses, General Idea of, . . , . 42 
City Houses, Arrangements for.. . 37 

Civil Architecture Defined 5 

Classic Architecture, Mouldings 

in 323 

Classic Moulding, Annulet or Fil- 
let 323 

Classic Moulding, Astragal or 

Bead 323 

Classic Moulding, Cavetto or 

Cove 323 

Classic Moulding, Cyma-Recta. . . 324 
Classic Moulding, Cyma-Reversa. 324 

Classic Moulding, Ogee . 324 

Classic Moulding, Ovolo 323 

Classic Moulding, Scotia 323 

Classic Moulding, Torus 323 

Coffer Walls 49 

Cohesive Strength of Materials. . . 76 

Collar Beam in Truss 238 

Cologne, Cathedral of 11 

Columns, Antique, Form of 48 

Column, Base, Shaft and Capital. 14 
Columns, Egyptian, Dimensions, 

etc 33 

Column, Gothic Pillar, Form of.. 48 

Column, Outline of. . . 47 

Columns, Parthenon at Athens, 

Forms of 48 

Column or Pillar 47 

Column, Resistance of 47 

Column, Shaft, Form of. 47 

Column, Shaft, Swell of, Called 

Entasis 48 

Complex, or Ground Vault 52 

Composite Arch of Titus 28 

Composite, Corinthian or Roman 
Order 28 



PAGE 

Compression, Resistance to 77 

Compression, Resistance to Crush- 
ing and Bending 85 

Compression, Resistance to, Pres- 
sures Classified 83 

Compression, Resistance to, Table 

of 79 

Compression, Resistance to, in 

Proportion to Depth loi 

Compression at Right Angles and 

Parallel to Length 206 

Compression of Stout Posts 89 

Compression and Tension, 

Framed Girders 174 

Compression Transversely to Fi- 
bres SO 

Cone Defined 548 

Conic Sections 584 

Conjugate Axis Defined 548 

Conjugate Diameters to Axes of 

Ellipse 487 

Construction Essential 56 

Construction of Floors, Roof, 

etc.. Economy Important 123 

Construction, Framing,' Heav}'^ 

Weight 56 

Construction, Joints, Effect of 

Many 123 

Construction, Object of Defined. . 123 
Construction, Simplest Form Best. 123 
Construction, Superfluous Mate- 
rial 56 

Contents, Table of, General. . .613-624 
Corinthian Capital, Fanciful Ori- 
gin of. 24 

Corinthian Order Appropriate in 

Buildings 24 

Corinthian Order, Character of.... 16 
Corinthian Order, Description of. 23 
Corinthian Order, Elegance of. . . . 23 
Corinthian Order, The Favorite 

at Rome 27 

Corinthian Order, Grecian Origin 

of 16 

Corinthian Order, Modification of. 27 
Cornice, Angle Bracket, To Ob- 
tain the 343 

Cornice, Eaves, To Find Depth of. 335 



662 



INDEX. 



PAGE 

Cornice, Mouldings, Depth of. . . . 342 

Cornice, Projection, To Find 342 

Cornice, Projecting Part of En- 
tablature 15 

Cornice, Rake and Level Mould- 
ings, To Match 344 

Cornice, Shading, Rule for 611 

Cornice, Stucco, for Interior, De- 
signs 340 

Corollary Defined (Geometry) 348 

Corollary of Triangle and Right 

Angle 355 

Cottage Style, English 35 

Country-Seat, Style of a 37 

Cross-Bridging, Additional 

Strength by , 137 

Cross-Bridging, Furring Impor- 
tant 137 

Cross-Bridging, Resistance of Ad- 
joining Beams 139 

Cross- or Herring-Bone Bridging 

Defined 137 

Cross-Furring Defined 125 

Cross-Strains, Resistance to. . . .77, 99 
Crushing and Bending Pressure.. 85 
Crushing, Liability of Rafter to. . . 205 
Crushing Strength of Stout Posts. 89 

Cube Root, Examples in 645 

Cubes, Squares and Roots, Table 

of 638-645 

Cubic Feet to Gallons, To Reduce. 653 

Cupola or Dome 53 

Curb or Mansard Roof 54 

Curve Ellipse, Equations to 482 

Curve Equilibrium of Dome 218 

Cylinder, Denned 549 

Cylinder, Platform Stairs 24S 

Cylinder, Platform Stairs, Lower 

Edge of 249 

Cylinders and Prisms, Stair-Build- 
ing 257 

Cyma-Recta, Classic Moulding... 324 
Cyma-Reversa, Classic Moulding. 324 
Cyma-Recta, Grecian Moulding . . 327 
Cyma-Reversa, Grecian Moulding. 328 

Deafening, Weight per Foot 177 

Decagon, Defined 546 



PAGE 

Decimals, Reduction of. Examples 647 
Decorated Style, 14th Century 11 

Decoration, Attention to be given 
to 46 

Decoration, Roman 27 

Deflection, Defined 112 

Deflection, Differs in Different Ma- 
terials 113 

Deflection, Elasticity not Dimin- 
ished by 112 

Deflection, Floor-beams, Dwell- 
ings, Dimensions 127 

Deflection, Floor-beams, First- 
class Stores, Dimensions 128 

Deflection, Floor-beams, Ordinary 
Stores, Dimensions 127 

Deflection, Lever, Principle of. . . 119 

Deflection, Lever and Beam, Rela- 
tion Between 119 

Deflection, Lever, To Find, Load 
at End 120 

Deflection, Lever, Breadth or 
Depth, Load at End 121 

Deflection, Lever, Load Uniform. 121 

Deflection, Lever, Breadth or 
Depth, Load Uniform 122 

Deflection, Lever, for Certain, 
Load Uniform 122 

Deflection, Load Uniform or at 
Middle, Proportion of 116 

Deflection, Load Uniform, Breadth 
and Depth 117 

Deflection, Load Uniform or at 
Middle, Proportion of 119 

Deflection, in Proportion to 
Weight 112 

Deflection, Resistance to. Rule 
for 113 

Deflection, Safe Weight for Pre- 
vention... no 

Deflection, Weight at Middle, 
Breadth and Depth 114 

Deflection, Weight at Middle, for 
Certain 114 

Deflection, Weight at Middle, Cer- 
tain, for ii5 

Deflection, Weight Uniform, for 
Certain 117 



INDEX. 



663 



PAGIt 

Deflection, Weight Uniform, Cer- 
tain, for 118 

Denominator, Least Common (Al- 
gebra) 404 

Denominator, Least Common 

(Fractions) 384 

Dentils, Teeth-like Mouldings in 

Cornice 20 

Diagonal Crossing Parallelogram 

(Geometry) 35 1 

Diagonal of Square Forming Oc- 
tagon 357 

Diagram of Forces, Example 166 

Diameter, Circle, Defined 547 

Diameter, Ellipse, Defined 549 

Diast)'le, Explanation of the Word 19 

Diastyle, Intercolumniation 20 

Diocletian, Baths, Splendor of 27 

Division, Fractions. Rule for 3S9 

Division, by Factors (Fractions).. 381 

Division, Quotient (Algebra) 419 

Division, Reduction (Algebra).,. 419 

Dodecagon, Defined 546 

Dodecagon, To Inscribe 569 

Dodecagon, Radius of Circles 

(Polygons) 452 

Dodecagon, Side and Area (Poly- 
gons) 453 

Dome, Abutments, Strength of. . . 53 

Domes of Cathedrals 53 

Dome, Character of 53 

Dome, Construction and Form 

of 216 

Dome, Construction and Strength 

of 53 

Dome, Cubic Parabola computed 219 

Dome or Cupola, the 53 

Dome, Curve of Equilibrium, rule 

for 218 

Dome, Halle du Bled, Paris 54 

Dome, Pantheon at Rome 53 

Dome, Pendentives of 53 

Dome, Polygonal, Shape of An- 
gle Rib 223 

Dome, Ribbed, Form and Con- 
struction 217 

Dome, Scantling for. Table of 
Thickness 218 



PAGE 

Dome, Small, over Stairways, Form 

of 220 

Dome, Spherical, To Form...... 221 

Dome, St. Paul's, London ..... 54 

Dome, Strains on. Tendencies of. 219 

Domes, Wooden 54 

Doors, Circular Head 320 

Doors, Circular Head, to Form 

Soffit 321 

Doors, Construction of 317 

Doors, Folding and Sliding, Pro- 
portions 316 

Doors, Front, Location of 320 

Doors, Height, Rule for. Width 

Given 315 

Door Hanging, Manner of 317 

Doors, Panel, Bottom and Lock 

Rail, Width 316 

Doors, Panel, Four Necessary... 317 
Doors, Panel, Mouldings, Width. 317 
Doors, Panel, Styles and Muntins, 

Width 316 

Doors, Panel, Top Rail, Width. .. 317 
Doors, Stop for. How to Form... 317 
Doors, Single and Double, height 

of 316 

Doors, Trimmings Explained. .. . 317 
Doors, Uses and Requirements of 315 

Doors, Width of 315 

Doors, Should not be Winding. .. 317 
Doors, Width and Height, Propor- 
tion of 315 

Doors, Width, Rule for, Height 

Given 316 

Doric Order, Character of 16 

Doric Order, Grecian Origin of. . 16 
Doric Order, Modified by the Ro- 
mans 27 

Doric Order, Used by Greeks only 

at First 19 

Doric Order, Peculiarities of 17 

Doric Order, Rudeness of 30 

Doric Order, Specimen Buildings 

in 19 

Doric Temples, Fanciful Origin 

of 17 

Doric Temples 19 

Drawing, Articles Required 536 



664 



INDEX. 



PAGE 

Drawing-board Better without 

Clamps 537 

Drawing-board Liable to Warp, 

How Remedied 537 

Drawing-board, Difficulty in 

Stretching Paper 539 

Drawing-board, Ordinary Size.... 536 
Drawing, Diagrams aid Under- 
standing 536 

Drawing, Inking in 542 

Drawing, Laying Out the 541 

Drawing, the Paper 537 

Drawing in Pencil, To Make Lines 542 
Drawing, Secure Paper to Board. 537 

Drawing, Shade Lining 543 

Drawing, Stretching Paper 537 

Durability in a Building. 37 

Dwelling, Arrangement of Rooms 38 
Dwellings, Floor-beams, To Find 

Dimensions 127 

Dwellings, Floor-beams, Safe 

Weight for 126 

Dwelling-houses, Dimensions and 
Style 37 

Eaves Cornice, Designs for 335 

Eaves Cornice, Rule for Depth... 335 
Ecclesiastical Architecture, Point- 
ed Style II 

Ecclesiastical Style, Origin of. . . . 14 

Echinus, Grecian Moulding 327 

Economy, Construction Floors, 

Roofs, Bridges 123 

Eddystone and Bell Rock Light 

House 48 

Egyptian Architecture 30 

Egyptian Architecture, Appropri- 
ate Buildings for 33 

Egyptian Architecture, Character 

of 33 

Egyptian Architecture, Origin in 

Caverns , 14 

Egyptian Architecture. Principal 

Features of 30 

Egyptian Columns, Dimensions 

and Proportions 33 

Egyptian Walls, Massiveness of. . 33 
Egyptian Works of Art 30 



PAGE 

Elasticity of Materials 84 

Elasticity not Diminished by De- 
flection 112 

Elasticity, Result of Exceeding 

Limit 120 

Elevation, a Front View 37 

Elevated Tie-beam Roof Truss 

Objectionable 214 

Ellipse, Area 48S 

Ellipse, Axes, Two, To Find, Di- 
ameter and Conjugate Given. .. 593 

Ellipse Defined 481 

Ellipse, Equations to the Curve. . 482 
Ellipse, Major and Minor Axes 

Defined 481 

Ellipse, Ordinates, Length of . . . 491 
Ellipse, Parameter and Axis, Re- 
lation of o 485 

Ellipse, Practical Suggestions. ... 489 
Ellipse, Semi-major, Axis Defined 486 

Ellipse, Subtangent Defined 486 

Ellipse, Tangent to Axes, Rela- 
tion of 485 

Ellipse, Tangent with Foci, Rela- 
tion of 487 

Ellipsis, Axes of, To Find (Geom- 
etry) 585 

Ellipsis, Conjugate Diameters (Ge- 
ometry) 593 

Ellipsis Defined 548, 585 

Ellipsis, Diameter Defined 549 

Ellipsis, Foci, To Find 586 

Ellipsis, by Intersecting Arcs 590 

Ellipsis, by Intersecting Lines... 588 

Ellipsis, by Ordinates 588 

Ellipsis, Point of Contact with 

Tangent, To Find 593 

Ellipsis, Proportionate Axes, to 

Describe with 594 

Ellipsis, Trammel, to Find, Axes 

Given 586 

Elliptical Arch, Joints, Direction 

of 233 

English Architecture, Early 11 

English Cottage Style Extensive- 
ly Used 35 

England and France, Fourteenth 
Century I2 



INDEX. 



66- 



PAGE j PAGE 

Entablature, above Columns and Flanges, Area of, Tubular Iron 



Horizontal 14 

Entasis, Swell of Shaft of Column 48 

Equal Angles Defined 349 

Equal Angles, Example in 350 

Equal Angles, in Circle 358 



Girder 155 

Flanges, Area of Bottom, Tubular 

Iron Girder 159 

Flanges, Load at Middle of Cast- 
iron Girder, Sizes 162 



Equal Angles (Geometry) 553 Flanges, Load Uniform on Tubu- 

Equilateral Rectangle, to De- ! lar Iron Girder, Sizes 156 

scribe 56S ! Flanges, Proportion of, Tubular 

Equilateral Triangle Defined (Ge- ] Iron Girder. ,. ." 157 

ometry) 545 Flexure, Compared with Rup- 

Equilateral Triangle, to Construct ture. . . 84 

(Geometry) 568 Flexureof Rafter 205 

Equilateral Triangle, to Describe j Flexure, Resistance to. Defined. . 145 

(Geometry) 566 j Floor-arches, How Constructed. . 153 

Eqilateral Triangle, to Inscribe i Floor-arches, Tie-rods, Dwellings, 

(Geometry) 5691 Sizes 153 



Equilateral Triangle (Polygons).. 445 

Eustyle Defined 20 

Exponents, Quantities with Nega- 
tive (Algebra) 423 

Extrados of an Arch 52 

Face Mould, Accuracy of. Wind- 
ing Stairs 295 

Face Mould, Curves Elliptical, 
Winding Stairs 301 

Face Mould, Drawing of, Winding 
Stairs 296 

Face Mould, Sliding of. Winding 
Stairs 299 

Face Mould, Application of, Plat- 
form Stairs 275 

Face Mould, a Simple, Kell's 
Method for 268 

Factors, Multiplication (Algebra) 409 

Factors, Two, Squaring Difference 
of (Algebra) 412 

Fibrous Structure of Materials. . . 76 

Figure Equal, Given Figure (Ge- 



Floor-arches, Tie-rods, First-class 
Stores, Sizes 153 

Floor-beams, Distance from Cen- 
tres, Sizes Fixed 129 

Floor-beams, Dwellings, Safe 
Weight for 126 

Floor-beams, Dwellings, Deflec- 
tion Given, Sizes 127 

Floor-beams, First-class Stores, 
Deflection Given, Sizes 128 

Floor-beams, Ordinary Stores, De- 
flection Given, Sizes 127 

Floor-beams, Stores, Safe Weight 
for 126 

Floor-beams, Reference to Rules 
for Sizes 125 

Floor-beams, Reference to Trans- 
verse Strains 126 

Floor-beams, Proportion of 
Weight on All 130 

Floors Constructed, Single or 
Double 124 

Floors, Fire-proof Iron, Action of 

ometry) 575 j Fire on 143 

Figure, Nearly Elliptical, To Make Floors, Framed, Seldom Used. . . 124 



(Geometry) 591 

Fillet or Amulet, Classic Mould- 



Floors, Framed, Openings in 130 

Floors, Headers, Defined 130 



ing. 323 Floors, Ordinary, Efl^ect of Fire 

Fire-proof Floors, Action of Fire on 143 

on 143 1 Floors, Solid Timber, Dwellings 

Flanges, Cast-iron Girder 1631 and Assembly, Depth 143 



666 



INDEX. 



PAGE 

Floors, Solid Timber, First-class 

Stores, Depth 144 

Floors, Solid Timber, to Make 

Fire-proof 143 

Floors, Tail-beams Defined 130 

Floors, Trimmers or Carriage- 
beams Defined 130 

Floors, Wooden, More Fire-proof 

than Iron, Some Cases 143 

Fl)'ers and Winders, Winding 

Stairs 251 

Foci Defined 548 

Foci of Ellipsis To Find 586 

Foci of Ellipse, Tangent 487 

Force Diagram, Load on Each 

Support 179 

Force Diagram, Truss, Figs. 59, 

68 and 69 179 

Force Diagram, Truss, Figs. 60, 70 

and 71 180 

Force Diagram, Truss, Figs. 61, 

72 and 73 181 

Force Diagram Truss, Figs. 63 

74 and 75 183 

Force Diagram, Truss, Figs. 64, 

77 and 78 184 

Force Diagram, Truss, Figs. 65, 

78 and 79 185 

Force Diagram, Truss, Figs. 66, 

80 and 81 186 

Forces, Parallelogram cf 59 

Forces, Composition of 66 

Forces, Composition, Reverse of 

Resolution 67 

Forces, Resolution of 59 

Forces, Resolution of. Oblique 

Pressure 59 

Foundations, Description of . . . . 47 
Foundations in Marshes, Timbers 

Used 47 

Fractions, Addition, Like Denom- 
inators 382 

Fractions Added and Subtracted 

(Algebra) 403 

Fractions Changed by Division.. 380 

Fractions Defined 378 

Fractions, Division, Rule for 389 

Fractions, Division by Factors. .. 381 



PAGE 

Fractions Divided Graphically... 388 

Fractions Graphically Expressed. 378 

Fractions, Improper, Defined.... 380 

Fractions, Least Common Denom- 
inator 384 

Fractions, Multiplication, Rule. . . 387 

Fractions Multiplied Graphically. 386 

Fractions, Numerator and Denom- 
inator 378 

Fractions, Reduce Mixed Num- 
bers.- 381 

Fractions, Reduction to Lowest 
Terms 384 

Fractions Subtracted (Algebra). . . 405 

Fractions, Subtraction Like De- 
nominators 383 

Fractions, Unlike Denominators 
Equalized 383 

Framed Beams, Thickness of, 
Rules 130 

Framed Girder, Bays Defined.... 167 

Framed Girders, Compression and 
Tension, Dimensions 174 

Framed Girders, Construction 
and Uses 166 

Framed Girders, Height and 
Depth 167 

Framed Girders, Kinds of Pres- 
sure 173 

Framed Girders, Long, Construc- 
tion of 174 

Framed Girders, Panels on Under 
Chord, Table of 167 

Framed Girders, Ties and Struts, 
Effect of 174 

Framed Girders, Triangular Pres- 
sure, Upper Chord 168 

Framed Girders, Triangular Pres- 
sure, Both Chords 171 

Framed Openings in Floors 130 

Framing Beams, Effect of Splic- 
ing 235 

Framing Roof Truss 237 

Framing Roof Truss, Iron Straps, 
Size of o 239 

France and England, Fourteenth 
Century 12 

Friction, Effect of 82 



INDEX. 



667 



PAGE 1 



Frieze between Architrave and 

Cornice 

Furring Defined 

Gable, a Pediment in Gothic Ar- 
chitecture 

Gaining a Beam Defined 

General Contents, Table of. . . .613- 

Geometrical Progression (Alge- 
bra) 

Geometry, Angles of Triangle, 
Three, Equal Right Angle 

Geometry Chords Giving Equal 
Rectangles 

Geometry Defined 

Geometry, Divide a Given Line. . 

Geometry, Divisions in Line Pro- 
portionate 

Geometry, Elementary 

Geometry, Equal Angles 

Geometry, Equal Angles, Ex- 
ample 

Geometry, Figure Equal to Given 
Figure, Construct 

Geometry, Figure Nearly Ellipti- 
cal by Compasses 

Geometry, Measure an Angle. . . . 

Geometry Necessary in Handrail- 
ing. Stairs 

Geometry, Opposite Angles Equal. 

Geometry, Parallel Lines 

Geometry, a Perpendicular, To 
Erect 

Geometry,Perpendicular,let Fall a. 

Geometry, Perpendicular, Erect at 
End of Line 

Geometry, Perpendicular, Let Fall 
Near End of Line. . . , , 

Geometr}', Plane Defined (Stairs). 

Geometry, Point of Contact 

Geometry, Points, Three Given, 
Find Fourth 

Geometry, Right Line Equal Cir- 
cumference 

Geometry, Right Lines, Propor- 
tion Between 

Geometry, Right Lines, Two 
Given, Find Third 



15 
125 



15 
100 
■624 

435 

354 

363 
544 



583 
347 

553 

350 

575 

591 
34S 

257 
354 

555 

550 
551 

551 

553 
257 

558 

559 
566 

584 
582 



Geometry, Square Equal Rec- 
tangle, To Make 5S1 

Geometry, Square Equal Given 
Squares, To make 577 

Geometry, Square Equal Triangle, 
To Make 582 

German or Romantic Style, Thir- 
teenth and Fourteenth Centuries. 11 

Girder, Bearings, Space Allowed 
for 141 

Girder, Bow-String, Cast-iron, 
Should not be Used 163 

Girder, Bow-String, Substitute for. 163 

Girder, Construction with Long 

Bearings 140 

555 i Girder, Cast-Iron, Load Uniform, 

Flanges 163 

Girder, Cast-Iron, Load at Middle, 
Flanges 162 

Girder, Cast-Iron, Proper Form 
of iCi 

Girder Defined, Position and Use 
of 140 

Girder, Different Supports for 140 

Girder, Dwellings, Sizes for 141 

Girder, Framed, Bays Defined... . 167 

Girders, Framed, Compression 
and Tension, Dimensions 174 

Girder, Framed, Construction of.. 140 

Girder, Framed, Construction and 
Uses 166 

Girder, Framed, Construction of 
Long 174 

Girder, Framed, Kinds of Pres- 
sure 173 

Girders, Framed, Height and 
Depth 167 

Girders, Framed, Panels on Under 
Chord, Table of 1C7 

Girders, Framed, Triangular Pres- 
sure Upper Chord lOS 

Girders, Framed, Triangular Pres- 
sure Both Chords 171 

Girders, Framed, and Tubular 
Iron 140 

Girders, First-Class Stores, Sizes 
for 141 

Girders, Sizes, To Obtain 141 



668 



INDEX. 



PAGE 

Girders, Strengthening, Manner 

of 140 

Girders, Supports, Length of. Rule. 157 
Girders, Tubular Iron, Construc- 
tion of 154 

Girders, Tubular Iron, Area of 

Flange, Load at Middle 154 

Girders, Tubular Iron, Area of 

Flange, Load at any Point 155 

Girders, Tubular Iron, Area of 

Flange, Load Uniform 156 

Girders, Tubular Iron, Dwellings, 

Area of Bottom Flange 159 

Girders, Tubular Iron, First-Class 

Stores, Area of Bottom Flange.. 160 
Girders, Tubular Iron, Rivets, Al- 
lowance for 157 

Girders, Tubular Iron, Flanges, 

Proportion of 157 

Girders, Tubular Iron, Shearing 

Strain I57 

Girders, Tubular Iron.Web, Thick- 
ness of 158 

Girders, Weakening, Manner of. . 140 
Girders, Wooden, Objectionable. 154 
Girders, Wooden, Supporting, 

Manner of 154 

Glossary of Terms 627-637 

Gothic Arches 51 

Gothic Buildings, Roofs of 55 

Gothic and Norman Roofs, Con- 
struction of 178 

Gothic Pillar, Form of 48 

Gothic Style, Characteristics of. . . 12 

Goths, Ruins Caused by 12 

Granular Structure of Materials. . 76 

Gravity, Centre of. Position 71 

Gravity, Centre of. Examples, and 

Rule for 71 

Grecian Architecture, History of. 6 

Grecian Art, Elegance of 27 

Grecian Moulding, Cyma-Recta. . 327 
Grecian Moulding, Cyma-Re- 

versa 328 

Grecian Moulding, Echinus and 

Cavelto 327 

Grecian Moulding, Scotia 326 

Grecian Moulding, Torus 326 



PACE 

Grecian Orders' Modified by the 

Romans 27 

Grecian Origin of the Doric Or 

der t6 

Grecian Origin of Ionic Order ... 16 

Grecian Style in America 13 

Grecian Styles, their Different 

Orders j5 

Greek Architecture, Doric Order 

Used iQ 

Greek Building 35 

Greek Moulding, Form of 325 

Greek, Persian, and Caryatides 

Orders 24 

Greek Style Originally in Wood.. 14 
Greek Styles Only Known by 

Them 16 

Groined or Complex Vault 52 

Halle du Bled, Paris, Dome of, . . 54 

Halls of Justice, N. Y. C, Speci- 
men of Egyptian Architecture. . 8 

Handrailing, Circular Stairs 278 

Handrailing, Platform Stairs. ... 269 

Handrailing, Platform Stairs, Face 
Mould 264 

Handrailing, Platform Stairs, 
Large Cylinder 271 

Handrailing Stairs, Geometry 
Necessary 257 

Handrailing Stairs, " Out of Wind" 
Defined 257 

Handrailing Stairs, Tools Used. . 257 

Handrailing, Winding Stairs. .256, 289 

Handrailing Winding Stairs, Bal- 
usters Under Scroll 310 

Handrailing, Winding Stairs, 
Centres in Square 308 

Handrailing, Winding Stairs, Face 
for Scroll 311 

Handrailing, Winding Stairs, Fall- 
ing Mould 310 

Handrailing, Winding Stairs, Gen- 
eral Considerations 258 

Handrailing, Winding Stairs, 
Scroll for 308 

Handrailing, Winding Stairs, 
Scroll at Newel 309 



INDEX. 



669 



PACE 

Handrailing, Winding Stairs, 

Scroll Over Curtail Step 309 

Handrailing, Winding Stairs, 

Scroll for Curtail Step 310 

Headers, Breadth of 130 

Headers Defined 130 

Headers, Mortises, Allowance for 

Weakening by 131 

Headers, Stores and Dwellings, 

Same for Both 132 

Hecadecagon, Complete Square 

(Polygons) 458 

Hecadecagon, Radius of Circles 

(Polygons) 455 

Hecadecagon, Rules (Polygons). . 459 
Hecadecagon, Side and Area 

(Polygons) 457 

Height and Projection, Numbers 

of an Order 16 

Hemlock, Weight per Foot Super- 
ficial 177 

Heptagon Defined 546 

Herring-bone Bridging Defined... 137 

Hexagon Defined 546 

Hexagon, To Inscribe 569 

Hexagons, Radius of Circles 447 

Hexastyle, Intercolumniation. . . . 20 
Hindoo Architecture, Ancient, 

Character of 30 

Hip-Rafter, Backing of 216 

Hip-Roofs, Diagram and Expla- 
nation 215 

History of Architecture. , 44 

Hogged Ridge in Roof Truss. . . . 238 
Homologous Triangles (Geom- 
etry) 362 

Homologous Triangles (Ratio 

and Proportion) 370 

Hooke's Theory of an Arch 50 

Hooke's Theory, Bridge Arch, 

Pressure on 51 

Hooke's Theory, Catenary Arch. . 51 
Horizontal and Inclined Roofing, 

Weight igo 

Horizontal Pressure on Roof, To 

Remove 74 

Horizontal Thrust in Beams 72 

Horizontal Thrust, Tendency of. . 88 



1>AGE 

Hut, Original Habitation 13 

Hydraulic Method, Testing 
Woods 80 

Hyperbola Defined 548, 585 

Hyperbola, Height, To Find, Base 
and Axis Given 585 

Hyperbola by Intersecting Lines. 595 

Hypothenuse, Equality of Squares 
(Algebra) 416 

Hypothenuse, Formula for (Trig- 
onometry) 516 

Hypothenuse, Side, To Find (Ge- 
ometr)^) 579 

Hypothenuse, Right Angled Tri- 
angle (Geometr}') 355 

Hypothenuse, Triangle (Trigo- 
nometry) 518 

Ichnographic Projection, Ground 

Plan 37 

Improper Fractions Defined 380 

India Ink in Drawing 540 

Inertia, Moment of. Defined 145 

Inking-in Drawing 542 

Inside Shutters for Windows, Re- 
quirements 319 

Instruments in Drawing 540 

Intercolumniation Defined 17 

Intercolumniation of Orders 20 

Intrados of Arch 52 

Ionic Order, Character of. 16 

Ionic Order, Grecian Origin of. . . 16 
Ionic Order Modified by the Ro- 
mans 27 

Ionic Order, Origin of 20 

Ionic Order, Suitable for What 

Buildings 20 

Ionic Volute, To Describe an. ... 20 
Iron Beams, Breaking Weight at 

Middle 148 

Iron Beams, Deflection, To Find, 

Weight at Middle 147 

Iron Beams, Deflection, To Find, 

Weight Uniform 150 

Iron Beams, Dimensions, To Find, 

Weight any Point 149 

Iron Beams, Dimensions, To Find, 

Weight Uniform 149 



670 



indp:x. 



PAGE 

Iron Beams, Dwellings, Distance 
from Centres 151 

Iron Beams, First-Class Stores, 
Distance from Centres 152 

Iron Beams, Rectangular Cross- 
Section , , 145 

Iron Beams, Rolled, Sizes 145 

Iron Beams, Safe Weight, Load 
any Point 148 

Iron Beams, Safe Weight, Load 
Uniform. .' 151 

Iron Beams, Table IV 146 

Iron Beams, Weight at Middle, 
Deflection Given 146 

Iron Fire-Proof Floors, Action of 
Fire On 143 

Iron Straps, Framing, to Prevent 
Rusting 239 

Irregular Polygon, Trigon (Geom- 
etry) 546 

Isosceles Triangle Defined 545, 584 

Italian Architecture, Thirteenth, 
Fourteenth, and Fifteenth Cen- 
turies 12 

Italian Use of Roman Styles 13 

Italy, Tuscan Order the Principal 
Style 30 

Jack-Rafters, Location of 212 

Jack-Rafters and Purlins in Roof. 211 
Jack-Rafters, Weight per Superfi- 
cial Foot 189 

Joists and Studs Defined 174 

Jupiter, Temple of, at Thebes, Ex- 
tent of 33 

Kell's Method, Simple Face 

Mould, Stairs 268 

Keystone for Arch, Position of . . . . 50 
King-Post, Bad Framing, Effect 

of 237 

King-Post, Location of 213 

King-Post in Roof. 54 

Lamina in Girders Defined 174 

Lancet Arch 51 

Lateral Thrust in Arch 52 



PAGE 

Laws of Pressure 57 

Laws of Pressure, Inclined, Ex- 
amples 57 

Laws of Pressure, Vertical, Exam- 
ples 57 

Leaning Tower or Campanile, 

Twelfth Century 12 

Length, Breadth, or Thickness, 

Relation to Pressure 78 

Lever, Breadth or Depth, To 

Find Ill 

Lever, Deflection as Relating to 

Beam 119 

Lever, Deflection, Load at End.. 120 

Lever, Deflection, Load Uniform. 121 
Lever, Deflection, Breadth or 

Depth, Load at End 121 

Lever, Deflection, Breadth or 

Depth, Load Uniform 122 

Lever, Deflection, Load Required. 122 
Lever Formula, Proportionals in 

(Algebra) 421 

Lever Load Uniformly Distrib- 
uted Ill 

Lever, Load at One End no 

Lever Principle Demonstrated 

(Ratio) 375 

Lever, Support, Relative Strength 

of One no 

Light-Houses, Eddystone and Bell 

Rock 48 

Line Defined (Geometry) 544 

Lines, Divisions in. Proportionate 

(Geometry) 583 

Lintel, Position of 49 

Lintel, Strength of 49 

Load, per foot, Horizontal 192 

Load on Roof Truss, per Superfi- 
cial Foot c 189 

Load on Tie-Beam, Ceiling, etc. . 190 

Lock Rail for Doors, Width 316 

Logarithms Explained (Algebra).. 425 

Logarithms, Examples 426 

Logarithms, Sine and Tangents 

(Polygons) 464 

Lombard, Byzantine Style 10 

Lombard Style, Seventh Century. 10 

London Bridge, Piles, Age of. . . . 229 



INDEX. 



671 



J'AGE 

Materials, Cohesive Strength of. . 76 
Materials, Compression, Resist- 
ance to 77 

Materials, Cross-strain, Resistance 

to 77 

Materials, Structure of 76 

Materials, Tension, Resistance to. 77 
Materials Tested, General De- 
scription 80 

Materials, Weights, Table of. . . . 654 
Major and Minor Axes of Ellipse 

Denned 4S1 

Marshes, Foundation for Timbers 

in 47 

Mathematics Essential 347 

Maxwell, Prof. I. Clerk, Diagrams 

of Forces, etc. , 165 

Memphis, Pyramids of, Estimate 

of Stone in 33 

Minster, Tower of Strassburg. ... 11 
Minutes, Sixty Equal Parts, to 

Proportion an Order 15 

Mixed Numbers in Fractions, To 

Reduce 381 

Modern Architecture, First Ap- 
pearance of 9 

Modern Tuscan, Appropriate for 

Buildings 30 

Moment of Inertia Defined 145 

Mono-triglyph, Explanation of the 

Word 19 

Monuments, Ancient, Their Archi- 
tects 6 

Moorish and Arabian Styles, An- 
tiquities of II 

Mortises, Proper Location of. • . . 100 
Mortising, Beam, Effect on 

Strength of 100 

Mortising Beam at Top, Injurious 

Effect of 100 

Mortising Beam, Effect of 231 

Mortising, Beam, Position of. . . . 236 
Mortising Headers, Allowance for 

Weakening 131 

Moulding, Classic, Astragal or 

Bead 323 

Moulding, Classic, Annulet or 
Fillet 323 



I PAGE 

Mouldings, Classic Architecture. 323 
Moulding, Classic, Cavctto or 

Cove 323 

Moulding, Classic, Cyma-Recta. . 324 

Moulding, Classic, C)'ma-Revcrsa. 324 

Moulding, Classic, Ogee 324 

Moulding, Classic, Ovolo 323 

oNIoulding, Classic, Scotia 323 

^loulding, Classic, Torus 323 

Mouldings, Common to all Or- 
ders 324 

Mouldings Defined 323 

Mouldings, Diagrams of. 330 

Mouldings, Doors, Rule for Width. 317 

Moulding; Grecian, Cyma-Rccta. 327 
Moulding, Grecian, Cyma-Re- 

versa 328 

Moulding, Grecian Echinus and 

Cavetto 327 

Mouldings, Greek, Form of. 325 

Mouldings, Grecian Torus and 

Scotia 326 

Mouldings, Modern 331 

Moulding, Modern, Antac Cap... 334 
Mouldings, Modern Interior, Dia- 
grams 332 

Mouldings, Modern, Plain 333 

Mouldings,Names, Derivations of. 324 

Mouldings,' Profile Defined 326 

Mouldings, Roman, Forms of. . . . 325 

Mouldings, Roman, Comments on. 329 
Mouldings, Roman, Ovolo and 

Cavetto 329 

Mouldings,Uses and Positions of. 324 

Multiplication (Algebra) 408 

Multiplication, Plus and Minus 

(Algebra) 415 

Multiplication, Three Factors (Al- 
gebra) 408 

Multiplication, Fractions 387 

Newel Cap, Form of. Winding 
Stairs 312 

Nicholson's Method, Plane 
Through Cylinder (Stairs) 259 

Nicholson's Method, Twists in 
Stairs 259 

Nonagon Defined 546 



6j: 



INDEX. 



PAGE 

Normal and Subnormal in Para- 
bola 496 

Norman and Gothic Construction 
of Roofs 17S 

Norman Style, Peculiarities of . . . 11 

Nosing and Tread, Position in 
Stairs 241 

Oblique Angle Defined 544 

Oblique Pressure, Resolution of 

Forces 59 

Oblique Triangle, Difference Two 

Angles (Trigonometr}^ 523 

Oblique Triangle, First Class 

(Trigonometry) 520 

Oblique Triangles, First Class, 

Formulae (Trigonometry) 531 

Oblique Triangles, Second Class 

(Trigonometr)') 522 

Oblique Triangles, Second Class, 

Formulae (Trigonometry) 532 

Oblique Triangles, Third Class 

(Trigonometry) 526 

Oblique Triangles, Third Class, 

Formulae (Trigonometry) 534 

Oblique Triangles, Fourth Class 

(Trigonometry) 528 

Oblique Triangles, Fourth Class, 

Formulae (Trigonometry) 534 

Oblique Triangles, Two Sides 

(Trigonometry) 521 

Oblique Triangles, Sines and 

Sides (Trigonometry) 519 

Obtuse Angle Defined 349, 544 

Obtuse Angled Triangle Defined. 545 
Octagon, Buttressed, Find Side 

(Geometry) 571 

Octagon Defined 546 

Octagon, Diagonal of Square 

Forming 357 

Octagon, Inscribe a (Geometry). . 570 

Octagon, Rules (Polygons) 451 

Octagon, Radius of Circles (Poly- 
gons) 449 

Octastyle, Intercolumniation 20 

Ogee Mouldings, Classic 324 

Opposite Angles Equal (Geome- 
try) 354 



Order of Architecture, Three 

Principal Parts 14 

Orders of Architecture, Persians 

and Caryatides 24 

Ordinates to an Arc (Geometry). . 563 

Ordinates, Circle, Rule for 471 

Ordinates of Ellipse 491 

Ostrogoths, Style of the 9 

Oval, To Describe a (Geometry). . 591 

Ovolo, Classic Moulding 323 

Ovolo, Roman Moulding 329 

Paper, The, in Drawing, Secure to 

Board 537 

Pantheon at Rome, Dome of, and 

Walls. 53 

Pantheon and Roman Buildings, 

Walls of 49 

Parabola, Arcs Described from... 503 

Parabola, Area, Rule for 509 

Parabola, Axis and Base, to find 

(Geometr)') 585 

Parabola, Curve, Equations to... . 493 

Parabola Defined 492 

Parabola Defined (Geometry).. 548, 585 

Parabola, Diameters 497 

Parabola Described from Ordi- 
nates 504 

Parabola Described from Diame- 
ters 507 

Parabola Described from Points.. 502 
Parabola of Dome Computed. . . . 219 

Parabola, General Rules 499 

Parabola by Intersecting Lines. . . 594 
Parabola Mechanically Described. 500 
Parabola, Normal and Subnor- 
mal 496 

Parabola, Ordinate Defined. .... 496 

Parabola, Subtangent. 496 

Parabola, Tangent 493 

Parabola, Vertical Tangent De- 
fined 495 

Parabolic Arch, Direction of 

Joints 234 

Parallel Lines Defined 544 

Parallel Lines (Geometry) 555 

Parallelogram, Construct a 576 

Parallelogram Defined 545 



INDEX. 



67. 



PAGE 

Parallelogram Equal to Triangles, 

To Make 576 

Parallelogram of Forces, Strains 

by 165 

Parallelograms Proportioned to 

Bases (Geometry) 360 

Parallelogram in Quadrangle 

(Geometry) 364 

Parallelogram, Same Base (Geom- 
etry) 352 

Parameter Defined 54S 

Parameter, Axes (Ellipse) 485 

Parthenon at Athens, Columns of. 48 
Partitions, Bracing and Trussing. 176 

Partitions, How Constructed 174 

Partition, Door in Middle, Con- 
struction 175 

Partition, Doors at End, Construc- 
tion of 176 

Partition, Great Strength, Con- 
struction 176 

Partitions, Location and Connec- 
tion 175 

Partitions, Materials, Quality of. . 175 
Partitions, Plastered, Proper Sup- 
ports for 175 

Partitions, Pressure on. Rules.... 177 
Partitions, Principal, of what Com- 
posed 175 

Partitions, Trussing in. Effects of. 175 
Pedestal, a Separate Substruc- 
ture 14 

Pediment, Triangular End of 

Building 15 

Pencil and Rulers, Drawing 540 

Pentagon Defined 546 

Pentagon, Circumscribed Circles 

(Polygons) 463 

Perpendicular Height of Roof, To 

find 579 

Perpendicular, Erect a 550 

Perpendicular, Erect a, at End of 

Line 551 

Perpendicular, Let Fall a 551 

Perpendicular, Let Fall a, at End 

of Line 553 

Perpendicular Style, Fifteenth 
Century 12 



Perpendicular in Triangle (Poly- 
gons) 440 

Persians, Origin and Description 

of 24 

Persians and Caryatides, Orders 

Used by Greeks 24 

Piers, Arrangement, in City Front 

of House 44 

Piers, Bridges, Construction and 

Sizes 228 

Piles, London Bridge, Age of. . . . 229 
Pine, White, Beams, Table of 

Weights for 177 

Pisa, Cathedral of. Eleventh Cen- 
tury 12 

Pisa, Cathedral of, Erection in 

1016 12 

Pise Wall of France 49 

Pitch Board, To Make, for Stairs. 247 

Pitch Board, Winding Stairs 252 

Plane Defined 257 

Plane Defined (Geometry) 544 

Plank,Weightof,on Roof, per foot. 189 
Plastering, Defective, To what 

Due 174 

Plastering, Strength of 174 

Plastering, Weight per foot 177 

Platform Stairs, Baluster, Posi- 
tion of 250 

Platform Stairs Beneficial 240 

Platform Stairs, Cylinder of. 248 

Platform Stairs, Cylinder, Lower 

Edge 249. 

Platform Stairs, Face Mould, Ap- 
plication of Plank 273 : 

Platform Stairs, Face Mould, 

Handrailing in 264 

Platform Stairs, Face Mould, Sim- 
ple Method 267 ■ 

Platform Stairs, Face Mould, 

Moulded Rails 274 

Platform Stairs, Face Mould, Ap- 
plication of 275 

Platform Stairs, Face Mould With- 
out Canting Plank 272- 

Platform Stairs, Handrail to 269 

Platform Stairs, Handrailing Large 
Cylinder 271 



674 



INDEX. 



PAGE 

Platform Stairs, Railing Where 

Rake Meets Level 272 

Platform Stairs, Twist-Rail, Cut- 
ting of 277 

Platform Stairs, Wreath of Round 

Rail 267 

Point of Contact (Geometry) 558 

Point Defined (Geometry) 544 

Pointed Style, Ecclesiastical Arch- 
itecture II 

Polygons, Angles of 462 

Polygons, Circumscribed and In- 
scribed Circles, Radius of 460 

Polygons Defined (Geometry) 546 

Polygons, Equilateral Triangle. . . 445 

Polygons, General Rules 461 

Polygons, Irregular, Trigon (Ge- 
ometry) 546 

Polygons, Perpendicular in Tri- 
angle 440 

Polygon, Regular, Defined (Geom- 
etry) 546 

Polygons, Regular, To Describe 

(Geometry) 573 

Polygons, Regular, To Inscribe in 

Circle (Geometry) 572 

Polygons, Sum and Difference, 

Two Lines 439 

Polygons, Table Explained 466 

Polygons, Table of Multipliers. . . 465 
Polygons, Triangle, Altitude of.. 442 
Polygonal Dome, Shape of Angle- 
Rib 223 

Posts, Area, To Find 86 

Posts, Diameter, To Find 92 

Posts, Rectangular, Safe Weight.. 92 
Posts, Rectangular,To Find Thick- 
ness 94 

Posts, Rectangular, Breadth Less 

than Thickness 96 

Posts, Rectangular, To Find 

Breadth 95 

Posts, To Find Side 93 

Posts, Slender, Safe Weight for. . 91 
Posts, Stout, Crushing Strength of. 89 

Pressures Classified 85 

Pressure, Oblique, Resolution of 
Forces 59 



PACK 

Pressure, Triangular, Framed 
Girders 171 

Pressure, Upper Chord, Triangu- 
lar Girder 168 

Prisms Cut by Oblique Plane 259 

Prisms and Cylinders, Stair-Build- 

i"g 257 

Prisms Defined (Stairs) 257, 259 

Prism, Top, Form of, in Perspec- 
tive 259 

Profile of Mouldings Defined 326 

Progression, Arithmetical (Alge- 
bra) 432 

Progression,Geometrical (Algebra) 435 
Projection and Height, Members 

of Orders of Architecture 16 

Protractor, Useful in Drawing... 541 
Purlins and Jack-Rafters in Roof. 211 

Purlins, Location of 212 

Pyramids of Memphis, Amount of 

Stone in 33 

Pycnostyle, Explanation of 20 

Quadrangle Defined 545 

Quadrangle Equal Triangle 353 

Quadrant Defined 547 

Quantities, Addition and Sub- 
traction (Algebra) 424 

Quantities, Division of (Algebra). 424 
Quantities, Multiplication of(Al. 

gebra) 424 

Queen-Post, Location of 213 

Queen-Post in Roof 54 

Radials of Rib in Bridge 226 

Radials of Rib for Wedges 226 

Radicals, Extraction of (Algebra). 425 

Radius of Arc, To Find 561 

Radius of Circle Defined 547 

Rafters, Braces, etc.. Length, To 

Find 580 

Rafters, Least Thrust, Rule for. . . 62 

Rafters, Length of, To Find 578 

Rafters, Liability to Crush Other 

Materials 205 

Rafters, Liability to Being Crushed 205 

Rafters, Liability to Flexure 205 

Rafters, Minimum Thrust of 62 



INDEX. 



6/5 



PAGE 

Rafters in Roof, EfTect of Weight 
on 179 

Rafters in Roof, Strains Subjected 
to 205 

Rafters and Tie-Beams, Safe 
Weight 87 

Rafters, Uses in Roof 54 

Rake in Cornice Matched with 
Level Mouldings 344 

Railing, Platform Stairs Rake 
Meets Level 272 

Ratio or Proportion, Equals Mul- 
tiplied 367 

Ratio or Proportion, Equality of 
Products 370 

Ratio or Proportion, Equality of 
Ratios 367 

Ratio or Proportion Equation, 
Form of 367 

Ratio or Proportion, Examples.. . 366 

Ratio or Proportion, Four Propor- 
tionals, to Find 377 

Ratio or Proportion, Homologous 
Triangles 370 

Ratio or Proportion, Lever Prin- 
ciple in 372 

Ratio or Proportion, Lever Prin- 
ciple Demonstrated 375 

Ratio or Proportion, Multiply an 
Equation 368 

Ratio or Proportion, Multiply and 
Divide One Number 368 

Ratio or Proportion, Rule of 
Three 366 

Ratio or Proportion, Steelyard as 
Example in 371 

Ratio or Proportion, Terms of 
Quantities 3C7 

Ratio or Proportion, Transfer a 
Factor 369 

Rectangle Defined 545 

Rectangle, Equilateral, To De- 
scribe 568 

Rectangular Cross-Section, Iron 
Beams 145 

Reduction Cubic Feet to Gallons, 
Rule 653 

Reduction Decimals, Examples.. 647 



Reflected Light, Opposite of 
Shade 611 

Regular Polygon in Circle, To In- 
scribe (Geometry) 572 

Regular Polygon Defined (Geom- 
etry) 54^ 

Regular Polygons, To Describe 
(Geometry) 573 

Resistance, Capability of 86 

Resistance to Compression, Ap- 
plication of Pressure 85 

Resistance to Compression, 
Crushing and Bending 85 

Resistance to Compression, Mate- 
rials 77 

Resistance to Compression, Pres- 
sure Classified 85 

Resistance to Compression in 
Proportion to Depth lor 

Resistance to Compression, Stout 
Posts, Rule 89 

Resistance to Compression, Table 
of Woods 79 

Resistance to Cross-Strains 77 

Resistance to Cross-Strains De- 
fined 99 

Resistance to Deflection, Rule.... 113 

Resistance Depending on Com- 
pactness and Cohesion 78 

Resistance Depending on Loca- 
tion, Soil, etc 79 

Resistance to Flexure Defined. . . 145 

Resistance Inversely in Propor- 
tion to Length 102 

Resistance to Oblique Force 206 

Resistance, Power of, How Ob- 
tained 78 

Resistance, Proportion to Area. . . 86 

Resistance, Strains, To What Due. 78 

Resistance to Tension Greatest 
in Direction of Length 8i 

Resistance to Tension, Proportion 
in Materials 81 

Resistance to Tension, Table of 
Materials 82 

Resistance to Tension, Materials. 77 

Resistance to Tension, Results 
from Trattsverse Strains 82 



676 



INDEX. 



Resistance to Transverse Strains, 

Table of 83 

Resistance to Transverse Strains, 

Description of Table 84 

Resistance Variable in One Ma- 
terial 79 

Reticulated Walls 49 

Rhomboid Defined 546 

Rhombus Defined 545 

Ribbed Bridge, Area of Rule .... 225 

Ribbed Bridge, Built 224 

Ribbed Bridge, Least Rise, Table 

of 224 

Right Angle Defined 348, 544 

Right Angle in Semicircle (Ge- 
ometry) 355 

Right Angle, To Trisect a 554 

Right Angled Triangle Defined . . 545 
Right Angled Triangle, Squares 

on (Algebra) 417 

Right Angled Triangles (Trigo- 
nometry) 510 

Right Angled Triangles, Formula 

for (Trigonometry) 530 j 

Right Lines (Geometry) 584 ; 

Right Line Equal Circumference. 566 
Right Lines, Mean Proportionals 

Between 584 

Right Lines, Two Given, Find 

Third 582 

Right Lines, Three Given, Find 

Fourth 5S3 

Right or Straight Line Defined. . . 544 

Right Prism Defined (Stairs) 257 

Risers, Number of. Rule to Ob- 
tain (Stairs) 246 

Rise and Tread (Stairs) 241 

Rise and Tread, Connection of 

(Stairs) 248 

Rise and Tread, Blondel's Method 

of Finding (Stairs) 242 

Rise and Tread, Table of, for 

Shops and Dwellings (Stairs). . . 245 
Rise and Tread, To Obtain (Wind- 
ing Stairs) 251 

Rolled Iron Beams, Extensive Use 

of 161 

Roman Architecture Defined 7 



Roman Architecture, Ruins of. . . 11 
Roman Architecture, Excess of 

Eniichm.ent 46 

Roman Building 26 

Roman Composite and Corinthian 

Orders 28 

Roman Decoration 27 

Roman Empire, Overthrow of 13 

Romans, Ionic Order Modified by 27 

Roman Moulding, Cavetto 329 

Roman Mouldings, Comments on. 329 

Roman Moulding, Ovolo 329 

Roman Mouldings, Forms of. 325 

Roman Pantheon, etc.. Walls of. . 49 
Roman Styles of Architecture. ... 26 
Roman Styles Spread by the Ital- 
ians 13 

Romantic or German Style, Thir- 
teenth and Fourteenth Centu- 
ries II 

Rome, Ancient Buildings of. 12 

Rome and Greece, Architecture 

of 8 

Roof, The 54 

Roofs, Ancient Norman and 

Gothic, Construction 178 

Roof Beams, Weight per Super- 
ficial Foot i8g 

Roof, Brace in, Rule Same as for 

Rafter 208 

Roofs. Construction of. 55 

Roof Covering, Mode of i83 

Roof Covering, Weights, Table of. 191 

Roof, Curb or Mansard 54 

Roofs, Diagrams and Description 

of 212 

Roof, Gothic Buildings 55 

Roofs, Gothic and Norman Build- 
ings, Construction 178 

Roofs, Hip, Diagram and Expla- 
nation 215 

Roof, Hip 54 

Roof, Horizontal Pressure, To Re- 
move from 74 

Roof, Jack-Rafters and Purlins.. . 211 

Roof, King-Post in 54 

Roof, Load per Foot Horizontal, 
Rule iq2 



INDEX. 



^77 



Roof, Load, Total per Foot Hori- 
zontal, Rule 197 

Roofs, Modern, Trussing Neces- 
sary 178 

Roofs, Norman and Gothic Build- 
ings 178 

Roof, Pent, To Find 54 

Roof, Perpendicular Height, To 

Find 579 

Roof Plank, Weight per Super- 
ficial Foot 189 

Roof, Planning a 188 

Roof, Pressure on 55 

Roof, Queen-Post in 54 

Roof, Rafters in 54 

Roof, Sagging, To Prevent 54 

Roof, Slope Should Vary Accord- 
ing to Climate, 191 

Roof Supports, Distance between. 189 
Roof, Suspension Rods, Safe 

Weight for 210 

Roof, Tie-Beam in 54 

Roof, Tie-Beam, Tensile Strain, 

Rule 204 

Roof Timbers, Mortising 55 

Roof Timbers, Scarfing of 55 

Roof Timbers, Splicing of 55 

Roof Timbers, Strains by Parallel- 
ogram of forces *. . . 198 

Roof Timbers, Strain Shown Ge- 
ometrically 199, 202 

Roof Truss, Arched Ceiling 214 

Roof Truss, Elevated Tie-Beam 

Objectionable 214 

Roof Truss, Elevating Tie-Beam, 

Effect of 187 

Roof Truss, Force Diagram, Figs. 

59, 68, and 69 179 

Roof Truss, Force Diagram, Figs. 

60, 70, and 71 180 

Roof Truss, Force Diagram, Figs. 

61, 72, and 73 181 

Roof Truss, Force Diagram, Figs. 

63, 74, and 75 183 

Roof Truss, Force Diagram, Figs. 

64, 77, and 78 184 

Roof Truss, Force Diagram, Figs. 

66, 80, and 81 186 



PAGE 

Roof Truss, Load on 189 

Roof Trusses, Strains, Effect of, on 

Different 179 

Roof Truss, Weights, Table of, per 

Superficial Foot 189 

Roof Truss, Weight per Superfi- 
cial Foot 190 

Roof, Trussing in 54 

Roof Trussing, Designs for 178 

Roof Trussing, Framing for 237 

Roof Trussing, Hogged Ridge.... 238 
Roof Trussing, King-Post, Effect 

of Bad Framing on 237 

Roofs, United States 55 

Roof, Vertical Pressure of Wind 

on, Effect of. 194 

Roof, Snow, Weight per Horizon- 
tal Foot 193 

Roof Weight on Rafter, Effect of.. 179 
Roof, Wind, Horizontal and Verti- 
cal Pressure of 193 

Roofing, Weight of Horizontal and 

Inclined 190 

Roofing, Weight per Superficial 

Foot 190 

Roots, Cubes, and Squares, Table 

of 638-645 

Round Post, Area of 90 

Rubble Walls 48 

Rulers and Pencil in Drawing... . 540 
Rupture Compared with Flexure. 84 
Rupture, Crushing, Safe Weight.. 89 

Rupture, Sliding, Safe Weight 87 

Rupture, Transverse, Safe Weight. 86 
Rusting Iron Framing Straps, To 
Prevent 239 

Safe Load for Material 81 

Safe Weight, Allowance for 84 

Safe Weight at Any Point, Rule. . 106 

Safe Weight, Beam at Middle 103 

Safe Weight, Bending 91 

Safe Weight, Beam, Breadth of, 

To Find 104 

Safe Weight, Beam, Depth, To 

Find 104 

Safe Weight, Breadth or Depth, To 

Find, Load at Middle 106 



6/8 



INDEX. 



PAGE 

Safe "Weight, Breadth or Depth, 

To Find, Load Uniform io8 

Safe Weight, Crushing Rupture. . 89 

Safe Weight Defined 84 

Safe Weight, Deflection, To Pre- 
vent no 

Safe Weight, Floor Beams, Dwell- 
ings 126 

Safe Weight, Floor Beams, Stores. 126 
Safe Weight, Load Evenly Distri- 
buted 107 

Safe Weight, Load per Superficial 

Foot of Floor 109 

Safe Weight, Margin Greater than 

Table 104 

Safe Weight, Rafter and Tie-Beam, 

Example 87 

Safe Weight, Rectangular Posts. . 92 

Safe Weight, Slender Posts 91 

Safe Weight, Sliding Rupture. ... 87 
Safe Weight, Strain at Middle of 

Beam 105 

Safe Weight, Suspension Rods in 

Roof 210 

Safe Weight, Tensile Strain 96 

Safe Weight, Tensile Strain, To 

Find 97 

Safe Weight, Transverse Rupture. 86 
Scale of Equal Parts, for an 

Order 15 

Scale, Use of, in Drawing 540 

Scalene Triangle Defined 545 

Scantling, Thickness for Domes. , 218 

Scotia, Classic Moulding 323 

Scotia, Grecian Moulding 326 

Sectional Area, Explanation of. . . 97 

Sector, Area in Circle 476 

Sector Defined 547 

Segment of Circle, Area, To Find. 479 

Segment of Circle Defined 547 

Segment of Circle, To Describe 

(Geometry) 560, 562 

Segment of Circle from Ordinates. 470 

Segment of Cylinder Defined 549 

Semi-major Axes of Ellipse De- 
fined 486 

Shade Lining, Drawing 543 

Shadow on Capital of Column.. . . 609 



PAGE 

Shadow of Circular Abacus on 

Column 608 

Shadow of Column on Wall 611 

Shadow on Cornice 611 

Shadow in Fireplace 605 

Shadow of Horizontal Beam 604 

Shadow, Inclination of the Line of. 596 
Shadow of Moulded Window Lin- 
tel 606 

Shadow of Nosing of Step 606 

Shadow of Pedestal on Steps 606 

Shadow of Projection on Cylindri- 
cal Wall 603 

Shadow in Recess 604 

Shadow in Recess, Back Vertical. 604 
Shadows, Reflected Light Opposite 

of 611 

Shadow and Shade, Distinction 

Between , 597 

Shadow of Shelf 598 

Shadow of Shelf, Acute Angled. . Coo 
Shadow of Shelf, Front Edge 

Curved 602 

Shadow of Shelf Inclined. ....... 600 

Shadow of Shelf Inclined in Verti- 
cal Section 601 

Shadow of Shelf on Inclined Wall. 603 
Shadow of Square Abacus on Col- 
umn 607 

Shadow on Straight Projections 

and Mouldings 587 

Shadow on Shelf of Uneven Width 599 

Shadows, Usefulness of 596 

Shaft, Base and Capital of Column 

Defined 14 

Shaft of Column 47 

Shaft, Upright Part of Column. . . 15 
Shearing Strain, Tubular Iron 

Girder I57 

Shutters, Inside, for Windows, 

Requirements 319 

Sines and Tangents, Logarithms 

(Polygons) 464 

Slate, Weight per Superficial Foot. 189 
Snow on Roof, Weight per Hori- 
zontal Foot 193 

Snow on Roof, Weight per Super- 
ficial Foot 190 



INDEX. 



679 



Soffit of Circular Headed Win- 

dows 321 

Solid Defined (Geometry) 544 

Solid Timber Floors, First-Class 

Stores, Depth 144 

Solid Timber Floors, Dwellings 

and Assembly, Depth 143 

Solid Timber Floors, Fire-Proof, 

To Make I43 

Span of Arch 52 

Spherical Dome, Shape, To Find. 221 

Splicing Beams 235 

Splicing, Depth of Indents, To 

Find 235 

Splicing, Effect of 236 

Splicing Roof Timbers 55 

Splicing or Scarfing Tie-Beam. . . . 234 

Spring in Arch. 52 

Square Defined 545 

Square or Cube Roots, Example. 645 
Squares, Cubes and Roots, Table 

of 638-645 

Square Equal Rectangle, To Make 

(Geometry) 581 

Square Equal Given Squares, To 

Make (Geometry) 581 

Square Equal Triangle, To Make 

(Geometry) 582 

Stability, Principles of 45 

Stairs, Circular, Face Mould, First 

Section 283 

Stairs, Circular, Face Mould for 

2S2, 2S5, 287 

Stairs, Circular, Falling Mould 

for Rail 281 

Stairs, Circular, Ilandrailing for. 27S 

Stairs, Circular, Plan of 279 

Stairs, Circular, Plumb Bevel De- 
fined 282 

Stairs, Circular, Timbers put in 

After Erection 253 

Stairs for Dwellings, Rule 244 

Stairs, Face Mould, Kell's Method 

for Simple 268 

Stairs, Handrailing, Geometry 

Necessary 257 

Stairs, Handrailing, "Out of 
Wind" Defined 257 



PAGE 

Stairs, Handrailing, Tools Used.. 257 

Stairs, Light and Ventilation 240 

Stairs, for Men and 'Women, Rise 

and Tread 243 

Stairs, Nosing and Tread, Posi- 
tion of 241 

Stairs, Pitch Board, To Make 247 

Stairs, Plan of. Defined 257 

Stairs, Plane and Cylinder, Nich- 
olson's Method 259 

Stairs, Platform, Baluster, Posi- 
tion of 250 

Stairs, Platform, Beneficial 240 

Stairs, Platform, Cutting Twist- 
Rail 277 

Stairs, Platform, Cylinder of 24S 

Stairs, Platform, Face IMould 

Without Canting Plank 272 

Stairs, Platform, Face Mould, Ap- 
plication to Plank 273 

Stairs, Platform, Face Mould, Ap- 
plication to Plank 275 

Stairs, Platform, Face INIould for 

Handrail 264 

Stairs, Platform, Face Mould, 

Moulded Rail 274 

Stairs, Platform, Face Mould, Sim- 
ple Method 267 

Stairs, Platform, Handrail to 269 

Stairs, Platform, Handrail, Large 

Cylinder 271 

Stairs, Platform, Handrail Where 

Rake Meets Level 272 

Stairs, Platform, Lower Edge of 

Cylinder 249 

Stairs, Platform, Wreath for Round 

Rail 267 

Stairs, Position and Require- 
ments 240 

Stairs, Prisms and Cylinders 257 

Stairs, Rise and Tread 241 

Stairs, Rise and Tread, Blondel's 

Method 242 

Stairs, Rise and Tread, Table for 

Shops and Dwellings 245 

Stairs, Rises, Number of. Height 

Given 246 

Stairs, Shops, Rise for 244 



680 



INDEX. 



PAGE 

Stairs, Space for Timber and Plas- 
ter 247 

Stairs, Stone, Public Building 240 

Stairs, String of, To Make 247 

Stairs, Tread, To Find, Rise Given 
242, 246 j 

Stairs, Tread and Riser Connec- 
tion. , 248 

Stairs, Width, Rule for 241 

Stairs, Winding, Balusters in 
Round Rail 313 

Stairs, Winding, Bevels in Splayed 
Work 314 

Stairs, Winding, Blocking Out 
Rail 301 I 

Stairs, Winding, Butt-joint on j 

Hand rail 303 

Stairs, Winding, Butt-joint, Cor- 
rect Lines for 307 

Stairs, Winding, Diagrams Ex- 
plained 263 

Stairs, Winding, Face Mould, Ac- 
curacy of 295 

Stairs, Winding, Face Mould, 
Application 297 

Stairs, Winding, Face Mould, 
Care in Drawing 295 

Stairs, Winding, Face Mould, 
Curves Elliptical 301 

Stairs, Winding, Face Mould 
for 290, 293 

Stairs, Winding, Face Mould, 
Round Rail 303 

Stairs, Winding, Face Mould for 
Twist 291 

Stairs, Winding, Flyers and 
Winders 251 

Stairs, Winding, Front String, 
Grade of 253 

Stairs, Winding, Handrailing.256, 289 

Stairs, Winding, Handrailing, Bal- 
usters Under Scroll 310 

Stairs. Winding, Handrailing, 

, Centres for Square 308 

Stairs, Winding, Handrailing, 
Face Mould for Scroll 311 

Stairs, Winding, Handrailing, Fall- 
ing Mould for Raking Scroll. . . 310 



Stairs, Winding, Handrailing, Gen- 
eral Considerations 258 

Stairs, Winding, Handrailing, 
Scrolls for 308 

Stairs, Winding, Handrailing, 
Scroll Over Curtail Step 309 

Stairs, Winding, Handrailing, 
Scroll for Curtail Step. 310 

Stairs, Winding, Scroll at Newel. 309 

Stairs, Winding, Illustrations by 
Planes 261 

Stairs, Winding, Moulds for 
Quarter Circle 255 

Stairs, Winding, Newel Cap, Form 
of 312 

Stairs, Winding, Objectionable. . 240 

Stairs, Winding, Pitch Board, To 
Obtain 252 

Stairs, Winding, Rise and Tread, 
To Obtain 251 

Stairs, Winding, Sliding of Face 
Mould 299 

Stairs, Winding, String, To Ob- 
tain 252 

Stairs, Winding, Timbers, Posi- 
tion of 252 

Stairs and Windows, How Ar- 
ranged 42 

Stiles of Windows, Allowance for. 319 

St. Mark, Tenth or Eleventh Cen- 
tury 12 

Stone Bridge Building, Truss 
Work 232 

Stone Bridge, Building Arch.... 230 

Stone Bridge, Centres for. Con- 
struction 229 

Stone Bridge, Pressure on Arch 
Stones 230 

Stop for Doors 317 

Stores, Floor Beams, Safe Weight. 126 

Stores, Ordinary, Floor-Beams, 
Sizes, To Find 127 

Stores, First-Class, Floor-Beams, 
Sizes, To Find 128 

St. Paul's, London, Dome of. . . . 54 

St. Peter's, Rome, Fourteenth and 
Fifteenth Centuries 12 

Straight or Right Line Defined. . . 544 



INDEX. 



68 1 



PAGE 

Strains, Cross, Resistance to 77 

Strains on Domes, Tendency of. . 219 

Strains Exceed Weights 61 

Strains, Graphic Representation.. 165 
Strain Greatest at Middle of Beam. 105 
Strains by Parallelogram of 

Forces 165 

Strains, Practical Method of De- 
termining 62 

Strains of Rafter in Roof 205 

Strains, Resistance, To What Due. 78 
Strain on Roof Timbers Shown 

Geometrically 190 

Strains on Roof Timbers Geomet- 
rically Applied 202 

Strains on Roof Timbers, Parallel- 
ogram of Forces. . 198 

Strain, Shearing, Tubular Iron 

Girder 157 

Strain Unequal, Cause of .... 83 

Straps, Iron, Roof Truss 239 

Strassburg, Cathedral of 12 

Strassburg, Towers of the Min- 
ster II 

Strength and Stiffness of Mate- 
rials 7S 

Structure of Materials 76 

Struts Defined 173 

Struts and Ties t)8 

Struts and Ties, Difference Be- 
tween 69 

St. Sophia, Sixth Century 12 

Stucco Cornice for Interior 340 

Studs and Joists Defined 174 

Styles, Grecian, Only Known by 

Them 16 

Stylobate, Substructure for Col- 
umns 14 

Subnormal and Normal (in Para- 
bola) 496 

Subtangent, Parabola 496 

Subtangent of Ellipse Defined. . . 486 
Subtraction and Addition (Alge- 
bra) ; 398 

Superficies Defined (Geometry)... 544 
Supports, Girders, Length, Rule.. 157 

Supports, Position of 65 

Supports, Inclination of, Unequal. 60 



Suspension Rods, Location in 

Roof.. 212 

Suspension Rods in Roof, Safe 

Weight 210 

Symbols Chosen at Pleasure (Al- 

I gebra) 395 

Symbols, Transferring (Algebra). 399 
1 Systyle, Explanation of 20 

i Table of Circles 649-652 

I Table of Contents 6x3-624 

i Table of Capacity of Wells, Cis- 

j terns, etc 653 

I Table of Squares, Cubes, and 

I Roots 638-645 

' Table of Woods, Description of.. 80 

Tail-Beams Defined 130 

Tanged Curve, To Describe (Ge- 
ometry) 565 

Tangent to Axes, Ellipse 485 

Tangent Defined 547 

I Tangent with Foci, Ellipse 4S7 

I Tangent to Ellipse, To Draw 592 

Tangent at Given Point in Cir- 

I cle 557 

Tangent at Given Point, Without 

Centre 557 

Tangent of Parabola 493 

Tangents and Sines, Logarithms 

(Polygons) 464 

Temples Built in the Doric Style. 19 

Temple, Doric, Origin of the 17 

Temple of Jupiter at Thebes 33 

Tenons and Splices, Knowledge 

Important 88 

Tensile Strain, Area of Piece, To 

Find 99 

Tensile Strain, Compressed Ma- 
terial 100 

j Tensile Strain, Condition of Sus- 
pended Piece 98 

Tensile Strain, Safe Weight 96 

Tensile Strain, Safe Weight, To 

Compute 97 

Tensile Strain, Sectional Area, To 

Obtain 97 

Tensile Strain, Suspended, Mate- 
terial Extended 100 



682 



INDEX. 



PACE 

Tensile Strain on Tie-Beam in Roof 

Truss 204 

Tensile Strain, Weight of Suspend- 
ed Piece 9S 

Tensile Strength of Cast Iron 161 

Tension and Compression, Fran:ied 

Girders 174 

Tension, Resistance to 77 

Tension, Resistance to, Table of 

Materials 82 

Tension, Resistance to. Results 

Obtained 82 

Tension, Resistance to. Proportion 

in Materials 81 

Tent, Habitation of the Shepherd. 13 
Testing Machine, Description in 

Transverse Strains 80 

Testing Materials, Hydraulic 

Method 80 

Testing Materials, Dates of 80 

Testing Materials, Manner of 80 

Tetragon Defined 546 

Tetragon, Radius of Circles 

(Polygons) 446 

Tetrastyle, Intercolumniation. ... 20 

Thebes, Thickness of Walls at. . . 33 

Thrust, Horizontal 63 

Thrust, Horizontal, Examples. ... 64 

Thrust, Horizontal, Tendency of.. 88 

Tie-Beam in Ceiling, Load on, . . 190 

Tie-Beam and Rafter, Safe Weight. 87 

Tie-Beam in Roof 54 

Tie-Beam in Roof, Tensile Strain. 204 
Tie-Rods, Diameter, To Find. . . . 164 
Tie-Rods, Floor Arches, Dwell- 
ings 153 

Tie-Reds, Floor Arches, First- 
Class Stores 153 

Tie-Rods, Wrought Iron 164 

Ties Defined 173 

Ties and Struts, To Distinguish.. 69 
Ties and Struts, Framed Girders.. 174 
Ties and Struts, Principles of. . . . 63 
Ties, Timbers in a State of Ten- 
sion 68 

Titus, Composite Arch of 28 

Trimmer, Breadth, To Find, Two 

Sets Tail-Beams 134 



TAGE 

Top Rail, Doors, Width, Rule 317 

Torus, Classic Moulding 323 

Torus, Grecian Moulding 326 

Tower of Babel, History of 5 

Towers of the Minster, Strassburg. ii 

Transverse Axis Defined 548 

Transverse Strains, Compressed 

and Extended, Material 100 

Transverse Strains, Defined 99 

Transverse Strains, Explanation 

of Table III loi 

Transverse Strains, Greater 

Strength of One Piece loi 

Transverse Strains, Neutral Line 

Defined 100 

Transverse Strains, Proportion to 

Bread th loi 

Transverse Slraiiis, Hatfield's, 

Reference to. .80, 121, 133,138, 

143, 144, 145, 146, 148 
Transverse Strains, Resistance to. 

Table of 83 

Transverse Strains, Description of 

Table 84 

Transverse Strains, Strength Di- 
minished by Division loi 

Trapezoid Defined 546 

Trapezium Defined 546 

Tread, To Find, Rise Given 

(Stairs) 242, 246 

Tread and Nosing, Position of 

(Stairs) 241 

Tread and Rise, To Find, Winding 

Stairs 251 

Tread and Rise, To Find, Blon- 

del's Method , 242 

Tread and Rise, Table for Shops 

and Dwellings 245 

Tread and Riser, Connection of 

(Stairs) 248 

Triangle, Altitude of (Poh'gons). 442 
Triangles, Base, Formula for (Trig- 
onometry) 516 

Triangle, Construct a (Geometry). 587 
Triangle, Construct Equal-Sided 

(Geometry) 575 

Triangle Defined 545 

Triangle, Examples (Geometry).. . 2>^ 



INDEX. 



683 



Triangles, Equal Altitude 361 

Triangle Equal Quadrangle 353 

Triangles, Equation of (Trigo- 
nometry) 515 

Triangles, Homologous (Geom- 
etry) 362 

Triangles, Hypothenuse, Formula 
for 5i(J 

Triangles, Hypothenuse, To Find 
(Trigonometry) 518 

Triangles, Perpendicular, To Find 
(Trigonometry) 517 

Triangle or Set-Square in Draw- 
ing 539 

Triangle or Set-Square, Use of. . . 541 

Triangles, Terms Defined (Trigo- 
nometry) 512 

Triangles, Three Angles Equal 
Right Angle 354 

Triangles, Value of Sides (Trigo- 
nometry) 516 

Trigon, Irregular Polygons (Ge- 
ometry) 546 

Trigon, Radius of Circle (Poly- 
gons) 443 

Trigon, Rule (Polygons) 441 

Trigonometry, Oblique Triangles, 
Two Angles 523 

Trigonometry, Oblique Triangles, 
Two Sides 521 

Trigonometry, Oblique Triangles, 
First Class 520 

Trigonometry, Oblique Triangles, 
Second Class 522 

Trigonometry, Oblique Triangles, 
Third Class 526 

Trigonometry, Oblique Triangles, 
Fourth Class 528 

Trigonometry, Oblique Triangles, 
Sines and Sides 519 

Trigonometry, Oblique Triangles, 
Formula, First Class 531 

Trigonometry, Oblique Triangles, 
Formula, Second Class 532 

Trigonometry, Oblique Triangles, 
Formula, Third Class 534 

Trigonometry, Oblique Triangles, j 

Formula, Fourth Class 534 i 



Trigonometry, Right Angled Tri- 
angles 510 

Trigonometry, Right Angled Tri- 
angles, Third Side, To Find... 511 

Trigonometry, Right Angled Tri- 
angle, Formula 530 

Trigonometry, Tables 513 

Trigonometry, Triangles, Base, 
Formula for 516 

Trigonometry, Triangles, Equa- 
tions of 515 

Trigonometry, Triangles, Hypoth- 
enuse, Formula 516 

Trigonometry, Triangles, Hypoth- 
enuse, To Find 518 

Trigonometry, Triangles, Perpen- 
dicular, To Find 517 

Trigonometry, Triangles, Terms 
Defined 512 

Trigonometry, Triangles, Value of 
Sides 516 

Trimmer or Carriage Beam, 
Breadth, To Find 132 

Trimmer or Carriage Beams De- 
fined 130 

Trimmer, One Header, Breadth, 
To Find, Dwellings and Stores. 133 

Trimmer, Well-Hole in Middle, 
Breadth, To Find 136 

Trisect a Right Angle 5154 

Truss, Diagram of 200 

Truss, Force Diagrams, Figs. 59, 

68 and 69 179 

Figs. 60, 70 and 71 iBo 

Figs. 61, 72 and 73 181 

Figs. 63, 74 and 75 183 

Figs. 64, 77 and 78 184 

Figs. 65, 78 and 79 185 

Figs. 66, 80 and 81 186 

Truss, Roof, Framing for 237 

Truss, Roof, Iron Straps 239 

Truss, Weight, per Horizontal 
Foot, To Find 192 

Truss Work, Stone Bridge Build- 
ing 232 

Trussing and Framing, Gravity 
and Resistance 76 

Trussing Partitions, Effect of.... 175 



684 



INDEX. 



PAGE 

Trussing Roofs, Effect of 178 

T-Square, How to Make 539 

Tubular Iron Girder, Area of Bot- 
tom Flange, Dwellings 159 

Tubular Iron Girder, Area of Bot- 
tom Flange, First-Class Stores. 160 
Tubular Iron Girder, Arc of 

Flange, Load at Middle 154 

Tubular Iron Girder, Area of 

Flange, Load Any Point 155 

Tubular Iron Girder, Area of 

Flange, Load Uniform 156 

Tubular Iron Girder, Flanges, 

Proportion of 157 

Tubular Iron Girder, Construction 

of 154 

Tubular Iron Girder, Rivets, Al- 
lowance for 157 

Tubular Iron Girder, Shearing 

Strain 157 

Tubular Iron Girders, Web of. . . 158 
Tuscan, Modern, Appropriate for 

Buildings 30 

Tuscan Order, Introduction of the. 30 
Tuscan Order, Principal Style in 

Italy 30 

Twelfth Century, Buildings in the. 11 

Twist Rail, Platform Stairs 277 

Twists, Stairs, Nicholson's Method 
for 259 

Undecagon Defined 546 

United States, Roofs in 55 

Vault, Simple, Groined or Com- 
plex 52 

Ventilation, Proper Arrangement 
for 45 

Versed Sine of Arc, To Find 561 

Vertical Pressure of Wind on 
Roof 194 

Vertical Tangent of Parabola De- 
fined 495 

Volutes, To Describe the 20 

Voussoir of an Arch 52 

Wall, The 48 

Walls, Coffer 49 



Walls, Construction and Forma- 
tion 48 

Walls, Eddystone and Bell Rock 

Lighthouses 48 

Walls, Egyptian, Massiveness of. 33 

Walls, Modern Brick 49 

Walls of Pantheon and Roman 

Buildings 49 

Walls of Pantheon at Rome 53 

Walls, Pise, of France 49 

Walls, Reticulated 49 

Walls, Rubble 48 

Walls, Strength of 48 

Walls, Various Kinds 49 

Walls, Wooden 49 

Weakening Girder, Manner of. . . 140 
Web of Tubular Iron Girder, 

Thickness of 15S 

Weight of Materials for Building 

Table of 654-656 

Wells, Cisteins, etc.. Table of 

Capacity 653 

White Pine, Weights of Beams 

Table of 177 

i Wind, Greatest Pressure, per Su- 
perficial Foot 90 

Wind on Roof, Effect of Vertical 

j Pressure 194 

Wind on Roof, Horizontal and 

( Vertical Pressure 193 

I Winders in Stairs, How to Place 

' the 42 

Winders and Flyers, Stairs 251 

I Windows, Arrangement of 44 

i Windows, Circular Headed 320 

Windows, Circular Headed, To 

Form Soffit 321 

Windows, Dimensions, To Find. 318 

Window-Frame, Size of 318 

Windows, Front of Building, Ef- 
fect of 320 

Windows, Heights, Table of, 

Width Given 320 

Windows, Height from Floor. . . . 320 
Windows, Inside Shutters, Re- 
quirement 319 

Windows, Position and Light 
from 317 



INDEX. 



685 



Windows and Stairs, How Ar- 
ranged 42 

Windows, Sliles, Allowance lor.. 3191 

Windows, Width Uniform, Height I 

Varying 319 

Winding Stairs, BaluGters in 
Round Rail 313 

Winding Stairs, Bevels in Splayed 
Work 314 

Winding Stairs, Blocking Out 
Rail 301 

Winding Stairs, Butt Joint, Posi- 
tion of. 303 

Winding Stairs, Butt Joint 307 

Winding Stairs, [Diagram of. Ex- 
plained 263 

Winding Stairs, Face Mould for 
290, 293 

Winding Stairs, Face Mould, Ac- 
curacy of. 295 

Winding Stairs, Face Mould, Ap- 
plication of 297 

Winding Stairs, Face Mould, 
Curves Elliptical 30T 

Winding Stairs, Face Mould, 
Drawing 296 

Winding Stairs, Face Mould, 
Round Rail .... 303 

Winding Stairs, Face Mould, Slid- 
ing of 299 

Winding Stairs, Face Mould for 
Twist 291 

Winding Stairs, Flyers and Wind- 
ers 251 

Winding Stairs, Front String, 
Grade of. 253 

Winding Stairs, Handrailing 
. c 256, 2S9 



Winding Stairs, Handrailing, oal- 
usters Under Scroll 310 

Winding Stairs, Handrailing, Cen- 
tres in Square 308 

Winding Stairs, Handrailing, Face 
Mould for Scroll 311 

Winding Stairs, Handrailing, Fall- 
ing Mould 310 

Winding Stairs, Handrailing, 
General Considerations 258 

Winding Stairs, Handrailing, 
Scrolls for 308 

Winding Stairs, Handrailing, 
Scroll Over Curtail Step 309 

Winding Stairs, Handrailing, 
Scroll for Curtail Step 310 

Winding Stairs, Handrailing, 
Scrolls at Newel 309 

Winding Stairs, Illustrations by 
Planes 261 

Winding Stairs, Moulds for Quar- 
ter Circle 255 

Winding Stairs, Newel Cap, Form 



of. 



312 

Winding Stairs Objectionable. . . . 240 
Winding Stairs, Pitch Board, To 

Obtain 252 

Winding Stairs, Rise and Tread, 

To Obtain 251 

Winding Stairs, String, To Obtain. 252 
Winding Stairs, Timbers, Posi- 
tion of. 252 

Wood, Destruction by Fire 37 

Wooden Beams, Use Limited 154 

Woods, Hydraulic Method of 

Testing 80 

Wreath for Round Rail, Platform 
Stairs 267 



THE END. 



DEC 4 - 1950 







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